
# 7.1E: Exercises

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## Verbal

1) We know $$g(x)=\cos x$$ is an even function, and $$f(x)=\sin x$$ and $$h(x)=\tan x$$are odd functions. What about $$G(x)=\cos ^2 x$$, $$F(x)=\sin ^2 x$$ and $$H(x)=\tan ^2 x$$? Are they even, odd, or neither? Why?

All three functions, $$F,G,$$ and $$H$$$,$ are even.

This is because

$$F(-x)=\sin(-x)\sin(-x)=(-\sin x)(-\sin x)=\sin^2 x=F(x),G(-x)=\cos(-x)\cos(-x)=\cos x\cos x= cos^2 x=H(-x)=\tan(-x)\tan(-x)=(-\tan x)(-\tan x)=\tan2x=H(x)$$

2) Examine the graph of $$f(x)=\sec x$$ on the interval $$[-\pi ,\pi ]$$$.$How can we tell whether the function is even or odd by only observing the graph of $$f(x)=\sec x$$?

3) After examining the reciprocal identity for $$\sec t$$$,$ explain why the function is undefined at certain points.

When $$\cos t = 0$$$,$ then $$\sec t = 10$$$,$ which is undefined.

4) All of the Pythagorean identities are related. Describe how to manipulate the equations to get from $$\sin^2t+\cos^2t=1$$ to the other forms.

## Algebraic

For the exercises 5-15, use the fundamental identities to fully simplify the expression.

5) $$\sin x \cos x \sec x$$

$$\sin x$$

6) $$\sin(-x)\cos(-x)\csc(-x)$$

7) $$\tan x\sin x+\sec x\cos^2x$$

$$\sec x$$

8) $$\csc x+\cos x\cot(-x)$$

9) $$\dfrac{\cot t+\tan t}{\sec (-t)}$$

$$\csc x$$

10) $$3\sin^3 t\csc t+\cos^2 t+2\cos(-t)\cos t$$

11) $$-\tan(-x)\cot(-x)$$

$$-1$$

12) $$\dfrac{-\sin (-x)\cos x\sec x\csc x\tan x}{\cot x}$$

13) $$\dfrac{1+\tan ^2\theta }{\csc ^2\theta }+\sin ^2\theta +\dfrac{1}{\sec ^\theta }$$

$$\sec^2 x$$

14) $$\left (\dfrac{\tan x}{\csc ^2 x}+\dfrac{\tan x}{\sec ^2 x} \right )\left (\dfrac{1+\tan x}{1+\cot x} \right )-\dfrac{1}{\cos ^2 x}$$

15) $$\dfrac{1-\cos ^2 x}{\tan ^2 x}+2\sin ^2 x$$

$$\sin^2 x+1$$

For the exercises 16-28, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

16) $$\dfrac{\tan x+\cot x}{\csc x}; \cos x$$

17) $$\dfrac{\sec x+\csc x}{1+\tan x}; \sin x$$

$$\dfrac{1}{\sin x}$$

18) $$\dfrac{\cos x}{1+\sin x}+\tan x; \cos x$$

19) $$\dfrac{1}{\sin x\cos x}-\cot x; \cot x$$

$$\dfrac{1}{\cot x}$$

20) $$\dfrac{1}{1-\cos x}-\dfrac{\cos x}{1+\cos x}; \csc x$$

21) $$(\sec x+\csc x)(\sin x+\cos x)-2-\cot x; \tan x$$

$$\tan x$$

22) $$\dfrac{1}{\csc x-\sin x}; \sec x$$ and $$\tan x$$

23) $$\dfrac{1-\sin x}{1+\sin x}-\dfrac{1+\sin x}{1-\sin x}; \sec x$$ and $$\tan x$$

$$-4\sec x \tan x$$

24) $$\tan x; \sec x$$

25) $$\sec x; \cot x$$

$$\pm \sqrt{\dfrac{1}{\cot ^2 x}+1}$$

26) $$\sec x; \sin x$$

27) $$\cot x; \sin x$$

$$\dfrac{\pm \sqrt{1-\sin ^2 x}}{\sin x}$$

28) $$\cot x; \csc x$$

For the exercises 29-33, verify the identity.

29) $$\cos x-\cos^3x=\cos x \sin^2 x$$

\begin{align*} \cos x-\cos^3x &= \cos x (1-\cos^2 x)\\ &= \cos x\sin ^x \end{align*}

30) $$\cos x(\tan x-\sec(-x))=\sin x-1$$

31) $$\dfrac{1+\sin ^2x}{\cos ^2 x}=\dfrac{1}{\cos ^2 x}+\dfrac{\sin ^2x}{\cos ^2 x}=1+2\tan ^2x$$

\begin{align*} \dfrac{1+\sin ^2x}{\cos ^2 x} &= \dfrac{1}{\cos ^2 x}+\dfrac{\sin ^2x}{\cos ^2 x}\\ &= \sec ^2x+\tan ^2x\\ &= \tan ^2x+1+\tan ^2x\\ &= 1+2\tan ^2x \end{align*}

32) $$(\sin x+\cos x)^2=1+2 \sin x\cos x$$

33) $$\cos^2x-\tan^2x=2-\sin^2x-\sec^2x$$

\begin{align*} \cos^2x-\tan^2x &= 1-\sin^2x-\left (\sec^2x -1 \right )\\ &= 1-\sin^2x-\sec^2x +1\\ &= 2-\sin^2x-\sec^2x \end{align*}

## Extensions

For the exercises 34-39, prove or disprove the identity.

34) $$\dfrac{1}{1+\cos x}-\dfrac{1}{1-\cos (-x)}=-2\cot x\csc x$$

35) $$\csc^2x(1+\sin^2x)=\cot^2x$$

False

36) $$\left (\dfrac{\sec ^2(-x)-\tan ^2x}{\tan x} \right )\left (\dfrac{2+2\tan x}{2+2\cot x} \right )-2\sin ^2x=\cos 2x$$

37) $$\dfrac{\tan x}{\sec x}\sin (-x)=\cos ^2x$$

False

38) $$\dfrac{\sec (-x)}{\tan x+\cot x}=-\sin (-x)$$

39) $$\dfrac{1+\sin x}{\cos x}=\dfrac{\cos x}{1+\sin (-x)}$$

Proved with negative and Pythagorean identities

For the exercises 40-, determine whether the identity is true or false. If false, find an appropriate equivalent expression.

40) $$\dfrac{\cos ^2 \theta -\sin ^2 \theta }{1-\tan ^\theta }=\sin ^2 \theta$$

41) $$3\sin^2\theta + 4\cos^2\theta =3+\cos^2\theta$$

\begin{align*} 3\sin^2\theta + 4\cos^2\theta &= 3\sin ^2\theta +3\cos ^2\theta +\cos^2\theta \\ &= 3\left ( \sin ^2\theta +\cos ^2\theta \right )+\cos^2\theta \\ &= 3+\cos^2\theta \end{align*}
42) $$\dfrac{\sec \theta +\tan \theta }{\cot \theta+\cos ^\theta }=\sec ^2 \theta$$