7.1E: Exercises

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?

Answer

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.

Answer

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

Answer

$\sin x$

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

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

Answer

$\sec x$

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

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

Answer

$\csc x$

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

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

Answer

$-1$

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

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

Answer

$\sec^2 x$

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

Answer

$\sin^2 x+1$

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

Answer

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

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

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

Answer

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

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

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

Answer

$\tan x$

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 $\tan x$

Answer

$-4\sec x \tan x$

24) $\tan x; \sec x$

25) $\sec x; \cot x$

Answer

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

26) $\sec x; \sin x$

27) $\cot x; \sin x$

Answer

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

28) $\cot x; \csc x$

For the exercises 29-33, verify the identity.

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

Answer

Answers will vary. Sample proof:

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

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

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

Answer

Answers will vary. Sample proof:

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

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

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

Answer

Answers will vary. Sample proof:

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

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

Answer

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

Answer

False

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

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

Answer

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

Answer

True

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

42) $\dfrac{\sec \theta +\tan \theta }{\cot \theta+\cos ^\theta }=\sec ^2 \theta$