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?

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