prove this qns plzz ​

Prove This Qns Plzz

Answers

Answer 1

Answer:

L.H.S.

= (cos5a.sin2a-cos4a.sin3a)/ (sin5a.sin2a-cos4a.cos3a)

Multiply numerator and denominator by 2.

= 2(cos5a.sin2a - cos4a.sin3a) / 2(sin5a.sin2a - cos4a.cos3a)

= (2cos5a.sin2a - 2cos4a.sin3a)/

(2sin5a.sin2a - 2cos4a.cos3a) = [sin(5a+2a)-sin(5a-2a)-sin(4a+3a)

+sin(4a-3a)]/[cos(5a-2a)-cos(5a+2a)-sin(4a-3a) +cos(4a+3a)]

= (sina - sin3a)/(cso3a-cosa)

= (-2cos2a.sina)/(-2sin2a.sina)

= cos2a/sin2a

= cot2a

= R.H.S.

Answer 2

answer is in a picture have a look

Prove This Qns Plzz

Related Questions

Tickets to a local movie were sold at $6.00 for adults and $4.50 for students. If 390 tickets were sold for a total of $2190.00, how many student tickets were sold

Answers

Answer: Therefore 100 student tickets were sold

Step-by-step explanation:

Let the number of student tickets be x

So adult tickets = 390 - x

ATQ

4.5(x) + 6(390-x) = 2190

4.5x + 2340 - 6x = 2190

-1.5x + 2340 = 2190

-1.5x = 2190-2340

-1.5x = -150

x = -150/-1.5

x = 100

Therefore 100 student tickets were sold

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The automatic opening device of a military cargo parachute has been designed to open when the parachute is 185 m above the ground. Suppose opening altitude actually has a normal distribution with mean value 185 and standard deviation 32 m. Equipment damage will occur if the parachute opens at an altitude of less than 100 m. What is the probability that there is equipment damage to the payload of at least one of five independently dropped parachutes

Answers

Answer:

0.0193 = 1.93% probability that there is equipment damage to the payload of at least one of five independently dropped parachutes.

Step-by-step explanation:

For each parachute, there are only two possible outcomes. Either there is damage, or there is not. The probability of there being damage on a parachute is independent of any other parachute, which means that the binomial probability distribution is used to solve this question.

To find the probability of damage on a parachute, the normal distribution is used.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Probability of a parachute having damage.

The opening altitude actually has a normal distribution with mean value 185 and standard deviation 32 m, which means that [tex]\mu = 185, \sigma = 32[/tex]

Equipment damage will occur if the parachute opens at an altitude of less than 100 m, which means that the probability of damage is the p-value of  Z when X = 100. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{100 - 185}{32}[/tex]

[tex]Z = -2.66[/tex]

[tex]Z = -2.66[/tex] has a p-value of 0.0039.

What is the probability that there is equipment damage to the payload of at least one of five independently dropped parachutes?

0.0039 probability of a parachute having damage, which means that [tex]p = 0.0039[/tex]

5 parachutes, which means that [tex]n = 5[/tex]

This probability is:

[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]

In which

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 0) = C_{5,0}.(0.0039)^{0}.(0.9961)^{5} = 0.9807[/tex]

Then

[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.9807 = 0.0193[/tex]

0.0193 = 1.93% probability that there is equipment damage to the payload of at least one of five independently dropped parachutes.

What's the next number in the sequence 16, 4, 1,

Answers

Answer:

0.25

Step-by-step explanation:

16/4 = 4

4/4 = 1

1/4 = 0.25

0.25/4 = 0.0625

0.0625/4 = 0.015625

give me brainliest please:)

the value of 5/121^1/2​

Answers

Answer:

√5/121

Step-by-step explanation:

formula: a^½=√a

(⁵/¹²¹)^½=√⁵/¹²¹

what expression is equivalent to (-7²-x-5)-(3x²+x)

Answers

Answer:

-3x² - 2x - 54

Step-by-step explanation:

(-7²-x-5)-(3x²+x)

-7² - x - 5 - 3x² - x

-49 - x - 5 - 3x² - x

-3x² - x - x - 49 - 5

-3x² - 2x - 54

if x and y are linear pair of angel then x +y=​

Answers

Answer: x + y = 180²

Step-by-step explanation:

A linear pair is a pair of adjacent, supplementary angles.

Adjacent means next to each other.

Supplementary means that the measures of the two angles add up to equal 180 degrees.

Therefore, by definition, if x and y are linear pairs of angles, then x + y = 180.

Aaron uses 18% of the paper in a printer paper package when she print a report for social studies class this is 90 sheets of paper how many sheets of paper are there in the printer paper in a package?

Answers

Answer:

percentage of paper used= 18%

Number of papers printed= 90

Total number of papers in the paper package= 'y'

total number of papers

= [tex]\frac{18}{100}[/tex] × y =90

transpose to the other side,

y= 90 ×[tex]\frac{100}{18}[/tex]

y=10 x100

y=1000

hence total number of papers in the package is 1000

Hope this helps

Please mark me as brainliest

Sphere A has a radius of 24 centimeters, and sphere B has a diameter of 42 centimeters. The radius of sphere A is
multiplied by what factor to produce the radius of sphere B?
A.4/7
B.7/8
C.8/7
D.7/4

Answers

Answer:

B

Step-by-step explanation:

If the diameter of sphere B is 42, that means the radius is half, so it is 21.

The question is asking what multiplied by 24 is 21, or 21 divided by 24. 21/24 can be simplified to 7/8.

Solve the equation for y. Identify the slope and y-intercept then graph the equation.

Y=-3x+1

Y=
M=
B=

Please Include a picture of the graph and show your work if you can

Answers

Answer:

y = 1        m = -3       b = 1

Step-by-step explanation:

y = mx + b                          y = -3x + 1

y = 1

m = -3

b = 1

(1,-19),(-2,-7) finding slope

Answers

Answer:

The slope is -4.

Step-by-step explanation:

Slope(m)=(y2-y1)/(x2-x1)

y2=-7, y1=-19, x2=-2, x1=1

(-7+19)/(-2-1)

=12/-3

=-4

Answer: -4

Step-by-step explanation:

The slope formula is: [tex]y_{2} -y_{1}/x_{2}-x_{1} \\[/tex]

So it is: (-7+19)/(-2-1) = 12/-3 = -4

I hope this helped!


What is the inverse of the function a(x)=1/x-2

Answers

Answer:

x = 1/x - 2

Step-by-step explanation:

Which equation is in standard form?
a. X +3= -5y
b. 5-y=x
c. y =3x + 6
d. -8x+ 3y = 12

Answers

选项 (d)

option (d)

!!!!!!!!!

I need help really bad

Answers

Answer:

1 ???????

Step-by-step explanation:

I need help on this plzzz I and not the best at math ​

Answers

Answer:

See attachment for graph

Step-by-step explanation:

Given

[tex]f(x) = \left[\begin{array}{cc}-1&x<-1\\0&-1\le x \le -1\\1&x>1\end{array}\right[/tex]

Required

The graph of the step function

Before plotting the graph, it should be noted that:

[tex]\le[/tex] and [tex]\ge[/tex] use closed circle at its end

[tex]<[/tex] and [tex]>[/tex] use open circle at its end

So, we have:

[tex]f(x) = -1,\ \ \ \ x < -1[/tex]

The line stops at -1 with an open circle      

[tex]f(x) = 0,\ \ \ \ -1 \le x \le 1[/tex]

The line starts at - 1 and stops at -1 with a closed circle at both ends

[tex]f(x) = 1,\ \ \ \ x > 1[/tex]

The line starts at 1 with an open circle

The options are not complete, so I will plot the graph myself.

See attachment for graph

Factorize:
625a^4 + 4b^4​

Answers

(625 • (a4)) + 22b4

54a4 + 22b4

Final result :

625a4 + 4b4

Power Function:
Consider the following graphs (1 and 2), and answer the questions FOR EACH GRAPH:

A) In what interval of the graph is it increasing, decreasing and constant? This answer must be justified by means of the definition
B) What is the domain and range?
C) Is it an odd or even function? This answer must be justified by means of the definition

Answers

Graph 1

Part (a)

The function is increasing when x > 0. The function is decreasing when x < 0.

The function is never constant

An increasing portion is when the graph goes uphill when moving left to right. A decreasing portion goes in the opposite direction: it goes downhill when moving left to right.

The reason why the function is never constant is because there aren't any flat horizontal sections. Such sections are when x changes but y does not. No such sections occur.

------------------------

Graph 1

Part (b)

Domain = set of all real numbers

Range = set of y values such that [tex]y \ge 0[/tex]

The domain is the set of all real numbers because we can plug in any value for x without any restriction. There are no division by zero errors to worry about, or square roots of negative numbers to worry about either.

The range is the set of nonnegative numbers as the graph indicates. The lowest y gets is y = 0.

------------------------

Graph 1

Part (c)

The function is even

The function f(x) = 1.6x^12 is an even function due to the even number exponent. For any polynomial, as long as the exponents are all even, then the function itself is even. If all the exponents were odd, then the function would be odd. This applies to polynomials only. A power function is a specific type of polynomial.

Note in the graph, we have y axis symmetry. The mirror line is vertical and placed along the y axis. This is a visual trait of any even function.

We could use algebra to show that f(-x) = f(x) like so

f(x) = 1.6x^12

f(-x) = 1.6(-x)^12

f(-x) = 1.6x^12

The third step is possible because (-x)^12 = x^12 for all real numbers x. It's similar to how (-x)^2 = x^2. You could think of it like (-1)^2 = (1)^2

============================================================

Graph 2

Part (a)

The function is decreasing when x < 0 and when x > 0

The function is never increasing

The function is never constant

In other words, the function is decreasing over the entire domain (see part b). The only time it's not decreasing is when x = 0.

The function is decreasing because the curve is going downhill when moving to the right. You can think of it like a roller coaster of sorts.

At no point of this curve goes uphill when moving to the right. Therefore, it is never increasing. The same idea applies to flat horizontal sections, so there are no constant intervals either.

------------------------

Graph 2

Part (b)

Domain: x is any real number but [tex]x \ne 0[/tex]

Range: y is any real number but [tex]y \ne 0[/tex]

Explanation: If we tried plugging x = 0 into the function, we get a division by zero error. This doesn't happen with any other number. Therefore, the set of allowed inputs is any number but 0.

The range is a similar story. There's no way to get y = 0 as an output.

If we plugged y = 0 into the equation, then we'd get this

y = 17x^(-3)

0 = 17/(x^3)

There's no way to have the right hand side turn into 0. The numerator is 17 and won't change. Only the denominator changes. We can't have the denominator be 0.  

------------------------

Graph 2

Part (c)

The function is odd

We can prove this by showing that f(-x) = -f(x)

f(x) = 17x^(-3)

f(-x) = 17(-x)^(-3)

f(-x) = 17* ( -(x)^(-3) )

f(-x) = -17x^(-3)

f(-x) = -f(x)

This is true for nearly all real numbers x, except we can't have x = 0.

Graphic 1:

(A) If f(x) = 1.6x ¹², then f '(x) = 19.2x ¹¹. Both f '(x) and x have the same sign, which means

• for -∞ < x < 0, we have f '(x) < 0, so that f(x) is decreasing on this interval

• for 0 < x < ∞, we have f '(x) > 0, so f(x) is increasing on this interval

f(x) is not constant anywhere on its domain.

(B) Speaking of domain, since f(x) is a polynomial (albeit only one term), it has

• a domain of all real numbers

• a range of {y ∈ ℝ : y = f(x) and y ≥ 0} (in other words, all real numbers y such that y = 1.6x ¹² and y is non-negative)

(C) This function is even, since

f(-x) = 1.6 (-x)¹² = (-1)¹² × 1.6x ¹² = 1.6x ¹² = f(x)

Graphic 2:

(A) Now if f(x) = 17/x ³, then f '(x) = -51/x ⁴. Because x ⁴ ≥ 0 for all x, this means f '(x) < 0 everywhere, except at x = 0. So f(x) is decreasing for (-∞ < x < 0) U (0 < x < ∞).

(B) f(x) has

• a domain of {x ∈ ℝ : x ≠ 0} (or all non-zero real numbers)

• a range of {y ∈ ℝ : y = f(x) and y ≠ 0} (also all non-zero reals)

(C) This function is odd:

f(-x) = 17/(-x)³ = 1/(-1)³ × 17/x ³ = -17/x ³ = -f(x)

4x+6=10. what is the value of x?

Answers

Answer:

1

Step-by-step explanation:

4x+6=10

4x=4

x=1

Answer:

[tex]4x + 6 = 10[/tex]

[tex]4x = 10 - 6[/tex]

[tex]4x = 4[/tex]

[tex]x = \frac{4}{4} [/tex]

[tex]x = 1[/tex]

hope this helps you

In a café, I order a cup of tea and a piece of cake and it costs £1.10. The next time I order 2 cups of tea and one piece of cake and it costs £1.70. Find the cost of a piece of cake.​

Answers

Answer:

£0.50

Step-by-step explanation:

t = one cup of tea

c = one piece of cake

t + c = £1.10

2t + c = £1.70

the cost increases by £0.60 (£1.70 - £1.10) when you order one more cup of tea which means that one cup of tea costs £0.60

substitute £0.60 into t + c = £1.10

£0.60 + c = £1.10

rearrange to get c = £1.10 - £0.60 = £0.50

so one piece of cake costs £0.50

Answer: piece of cake is 0.5£

Explanation:

Tea + cake = 1.10
2 Tea + cake = 1.70

From that we can find the value of a cup of tea by simply 1.70 - 1.10 = 0.6£

If cup of tea = 0.6£

Then a cake = 1.10 - 0.6 = 0.5 £

5. Given a test in which there is overlap of the test results for diseased and non-diseased individuals (e.g., normal individuals are found who have test results ranging in value from 8 to 15, and diseased individuals are found who have test results ranging in value from 12 to 25, so that in the range of values 12 to 15 there are both normal and diseased individuals), if the current cutoff value lies in the range of this overlap and you move the cutoff value toward the normal population (lower numbers in this example), the true negative numbers will _____________________ . (5 points)

Answers

Answer:

True negative numbers are considered as diseased individual. So, the true negative numbers will increase

Step-by-step explanation:

True negative numbers are considered as diseased individual. So, the true negative numbers will increase.

If 2m−6=8m
2
m
-
6
=
8
m
then 3m=
3
m
=


A. 3
B. -1
C. -3
D. -6
E. I don't know.

Answers

Answer:

option A

Step-by-step explanation:

please mark this answer as brainlist

Which of the following best describes the line that divides a design so that
every point on one side of the line coincides with a point on the other side of
the line?
A. Line of Symmetry
B. Point of Translation
C. Angle of Symmetry
D. Point of congruency

Answers

Answer:

Line of Symmetry i think

Line of symmetry best describes the line that divides a design so that every point on one side of the line coincides with a point on the other side of the line.

What is Coordinate Geometry?

Coordinate Geometry (or the analytic geometry) describes the link between geometry and algebra through graphs involving curves and lines.

A line of symmetry is a line that divides a figure into two congruent parts such that if one part is folded over the line of symmetry, it will coincide with the other part.

In other words, each point on one side of the line of symmetry is equidistant from the line as the corresponding point on the other side of the line.

The line that divides a design so that every point on one side of the line coincides with a point on the other side of the line is called the Line of Symmetry.

Hence,  line of symmetry best describes the line that divides a design so that every point on one side of the line coincides with a point on the other side of the line.

To learn more on Coordinate Geometry click:

https://brainly.com/question/18269861

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Max needs to paint a wall that is shaped like a square. He knows that the area of the wall is 75 ft2 . He needs to find the height of the wall. Find the height of the wall to the nearest tenth of a foot.

Answers

Answer:

8.7 feet

Step-by-step explanation:

Use the square area formula, a = s², where s is the side length of the square.

Plug in the area and solve for s:

a = s²

75 = s²

√75 = s

8.7 = s

So, to the nearest tenth of a foot, the height is 8.7 feet

the area of an equilateral triangle of side 8cm is
pls i need answer ASAP
I'll mark brainliest for anyone who can help me​

Answers

Side=a=8cm

[tex]\\ \sf\longmapsto Area=\dfrac{\sqrt{3}}{4}a^2[/tex]

[tex]\\ \sf\longmapsto Area=\dfrac{\sqrt{3}}{4}(8)^2[/tex]

[tex]\\ \sf\longmapsto Area=\dfrac{64\sqrt{3}}{4}[/tex]

[tex]\\ \sf\longmapsto Area=16\sqrt{3}[/tex]

[tex]\\ \sf\longmapsto Area=16\times 1.732[/tex]

[tex]\\ \sf\longmapsto Area=27.7cm^2[/tex]

11 10 Find the area of the shaded region. Round your answer to the nearest tenth.​

Answers

the answer is 110 because 11 times 10 is 110

What is the variable used in the equation 5x + 2 =100?​

Answers

Answer:

[tex]5x + 2 = 100 \\ 5x = 100 - 2 \\ 5x = 98 \\ x = \frac{98}{5} \\ x = 19.6[/tex]

Answer: the answer would be x because that's the actual variable in the question then if 19.6 was not an option

Step-by-step explanation:

Write the equation of a line in slope intercept form that passes through the two points. (-1,3) and (2,9)

Answers

Answer:

[tex]y = 2x +5[/tex]

Step-by-step explanation:

Finding the Slope:

m = rise/run

[tex]m=\frac{9-3}{2-(-1)}=\frac{6}{3}=\boxed{2}[/tex]

The slope is 2.

Finding the y-intercept:

[tex]y = 2x + b\\\\9 = 2(2) + b\\\\9 = 4 + b\\\\9 - 4 = 4 - 4 + b\\\\\boxed{ 5 = b}[/tex]

The y-intercept is (0,5).

The equation should be: [tex]y = 2x +5[/tex].

Hope this helps.

Answer:

y = 2x+5

Step-by-step explanation:

To find the equation in slope intercept form, we first have to find the slope of the two points.

The formula for finding the slope is:

[tex]\frac{y2-y1}{x2-x1}[/tex]

We are given the points:

(-1, 3) and (2, 9)

x1, y1 and x2, y2.

[tex]\frac{9-3}{2-(-1)}[/tex]

[tex]\frac{6}{3}[/tex] or 2 (the slope).

The slope intercept form is written as:

y= mx + b

y=2x+b

To find b, we can plug in either of the points given. Personally, I like to work with positive numbers. I'll be using the point (2, 9), however, either point will get you to the same answer.

y=2x+b

9=2(2)+b

9=4+b

Now subtract 4 from both sides.

5=b

Which then leaves us with the equation:

y = 2x+5

For a standard normal distribution, find:

P(z > c) = 0.058

Find c.

Answers

Answer:

1.572

Step-by-step explanation:

For a standard normal distribution,

P(z > c) = 0.058

To obtain C ; we find the Zscore corresponding to the proportion given, which is to the right of the distribution ;

Using technology or table,

Zscore equivalent to P(Z > c) = 0.058 is 1.572

Hence, c = 1.572

what should be added to 4x get 9X please help me in this pic also all​

Answers

Answer:

[tex]thank \: you[/tex]

What is the distance between the following points?
WILL GIVE BRAINLIEST

Answers

Answer:

D.√85

Step-by-step explanation:

We can find the distance between two points using the distance between two points formula

Distance between two points formula:

d = √(x2 - x1)² + (y2 - y1)²

Where the x and y values are derived from the given points

We are given the two points (-2,7) and (7,9)

Using these points let's define the variables ( variables are x1, x2, y1, and y2)

Remember points are written as follows (x,y)

The x value of the first point is -2 so x1 = -2

The x value of the second point is 7 so x2 = 7

The y value of the first point is 7 so y1 = 7

The y value of the second point is 9 so y2 = 9

Now that we have defined each variable let's find the distance between the two points

We can do this by substituting the values into the formula

Formula: d = √(x2 - x1)² + (y2 - y1)²

Variables: x1 = -2, x2 = 7, y1 = 7, y2 = 9

Substitute values in formula

d = √(7 - (-2))² + (9 - 7)²

Evaluation:

The two negative signs cancel out on 7-(-2) and it changes to +7

d = √ (7+2)² + (9-7)²

Add 7+2 and subtract 9 and 7

d = √ (9)² + (2)²

Simplify exponents 9² = 81 and 2² = 4

We then have d = √ 81 + 4

Finally we add 81 and 4

We get that the distance between the two points is √85

A route up a mountain is 20 Km long. john followed this route at an average speed of xkm/h. write down an expression in terms of x,for the number of hours he took to walk up the mountain.​

Answers

Answer:

20/x

Step-by-step explanation:

speed = distance /time

x km/h is speed

20 km is distance

x= 20/t

t= 20/x

Speed = d/t
T = d/ speed
T = 20/ X
Other Questions
I am having trouble converting :( A climate zone is an area that is defined by two main factors. What are they? Temperature and weather Precipitation and climate Weather and climate Temperature and precipitation Today I went to the zoo. I saw a(n) (adjective) (Noun) jumping up and down in its tree. He (verb, past tense) _(adverb) through the large tunnel that led to its (adjective) (noun). I got some peanuts and passed them through the cage to a gigantic gray _(noun) towering above my head. Feeding that animal made me hungry. I went to get a _(adjective) Scoop of ice cream. It filled my stomach. Afterwards I had to (verb) (adverb) to catch our bus. When I got home! (verb, past tense) my mom for a (adjective) day at the zoo. What does the word father of me Ivan is playing a skee-ball game. Different points are awarded depending on which hole the ball goes through. When the ball goes in the smallest hole, it is worth 100 points. When it goes in the bigger hole, it is worth 10 points, and when it does not go in either hole, it is worth 1 point. Ivan earned 352 points in the last game.Which combination will result in a score greater than his current score?2 balls in the smallest hole, and 8 balls in the bigger hole4 balls in the smallest hole, and 6 balls in neither hole3 balls in the smallest hole, 4 balls in the bigger hole, and 3 balls in neither hole3 balls in the smallest hole, 3 balls in the bigger hole, and 4 balls in neither hole A rectangular floor of area 360 m2 is going to be tiled. Each tile is rectangular, and has an area of 240 cm2. An exact number of tiles can be put into the space. How many tiles will be needed?? A car travels at a speed of x miles per hour for 3 hours and at half that speed for 2 hours. Which expression gives the total distance the car traveled in 5 hours? who many people in world What is the famous temple of Nepal.1.Pshupatinath Teple2.Swayambhunath3. Given two character strings s1 and s2. Write a Pthread program to find out the number of substrings, in string s1, that is exactly the same as s2 n 9What should you check to decidea speed is safe for your boat?Aoperator's age and experienceBwind and water conditionsvessel's capacity plateDspeed of other vessels Qu har Nancy en Panam?She will swim all week.She will go on a tour of Panama.She will go shopping.She will translate at a conference. solve the inequality x(x+6) >16 please show steps and interval notation! - Elabora un slogan que defina lo que es la ciencia y la investigacin How can COUNTIF, COUNTIFS, COUNT, COUNTA, and COUNTBLANK functions aide a company to support its mission or goals How long will it take money to double if it is invested at 25 % compounded continuously ? A car accelerates at 2 meters/s/s. Assuming the car starts from rest how far will it travel in 10 seconds Find the inverse mapping of x 2x +1 What are the first steps that you would take in planning an HCV intervention ? a. A breakdown of U.S. household garbage reveals the largest amount percentage-wise is paper and paperboard. b. In U.S. households most of the water usage is in the bathroom. c. The founder of the modern environmental movement and author of Silent Spring is Rachel Carson d. Deciduous trees such as most pines do not lose their leaves/foliage in winter.