Chapter 7

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?

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 \(f(x)=\sec x ?\)

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

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

Use the fundamental identities to fully simplify the expression. $$\sin x \cos x \sec x$$

Use the fundamental identities to fully simplify the expression. $$\sin (-x) \cos (-x) \csc (-x)$$

Use the fundamental identities to fully simplify the expression. $$\tan x \sin x+\sec x \cos ^{2} x$$

Use the fundamental identities to fully simplify the expression. $$\csc x+\cos x \cot (-x)$$

Use the fundamental identities to fully simplify the expression. $$\frac{\cot t+\tan t}{\sec (-t)}$$

Use the fundamental identities to fully simplify the expression. $$3 \sin ^{3} t \csc t+\cos ^{2} t+2 \cos (-t) \cos t$$

Use the fundamental identities to fully simplify the expression. $$-\tan (-x) \cot (-x)$$

Use the fundamental identities to fully simplify the expression. $$\frac{-\sin (-x) \cos x \sec x \csc x \tan x}{\cot x}$$

Use the fundamental identities to fully simplify the expression. $$\frac{1+\tan ^{2} \theta}{\csc ^{2} \theta}+\sin ^{2} \theta+\frac{1}{\sec ^{2} \theta}$$

Use the fundamental identities to fully simplify the expression. $$\left(\frac{\tan x}{\csc ^{2} x}+\frac{\tan x}{\sec ^{2} x}\right)\left(\frac{1+\tan x}{1+\cot x}\right)-\frac{1}{\cos ^{2} x}$$

Use the fundamental identities to fully simplify the expression. $$\frac{1-\cos ^{2} x}{\tan ^{2} x}+2 \sin ^{2} x$$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$\frac{\tan x+\cot x}{\csc x} ; \cos x$$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$\frac{\sec x+\csc x}{1+\tan x} ; \sin x$$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$\frac{1}{\sin x \cos x}-\cot x ; \cot x$$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$\frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x} ; \csc x$$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$(\sec x+\csc x)(\sin x+\cos x)-2-\cot x ; \tan x$$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$\frac{1}{\csc x-\sin x} ; \sec x \text { and } \tan x$$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$\frac{1-\sin x}{1+\sin x}-\frac{1+\sin x}{1-\sin x} ; \sec x \text { and } \tan x$$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$\tan x ; \sec x$$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$\sec x ; \cot x$$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$\cot x ; \sin x$$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$\cot x ; \csc x$$

Verify the identity. $$\cos x-\cos ^{3} x=\cos x \sin ^{2} x$$

Verify the identity. $$\cos x \tan x-\sec (-x) )=\sin x-1$$

Verify the identity. $$(\sin x+\cos x)^{2}=1+2 \sin x \cos x$$

Verify the identity. $$\cos ^{2} x-\tan ^{2} x=2-\sin ^{2} x-\sec ^{2} x$$

Prove or disprove the identity. $$\frac{1}{1+\cos x}-\frac{1}{1-\cos (-x)}=-2 \cot x \csc x$$

Prove or disprove the identity. $$\csc ^{2} x\left(1+\sin ^{2} x\right)=\cot ^{2} x$$

Prove or disprove the identity. $$\left(\frac{\sec ^{2}(-x)-\tan ^{2} x}{\tan x}\right)\left(\frac{2+2 \tan x}{2+2 \cot x}\right)-2 \sin ^{2} x=\cos 2 x$$

Prove or disprove the identity. $$\frac{\tan x}{\sec x} \sin (-x)=\cos ^{2} x$$

Prove or disprove the identity. $$\frac{\sec (-x)}{\tan x+\cot x}=-\sin (-x)$$

Prove or disprove the identity. $$\frac{1+\sin x}{\cos x}=\frac{\cos x}{1+\sin (-x)}$$

Determine whether the identity is true or false. If false, find an appropriate equivalent expression. $$\frac{\cos ^{2} \theta-\sin ^{2} \theta}{1-\tan ^{2} \theta}=\sin ^{2} \theta$$

Determine whether the identity is true or false. If false, find an appropriate equivalent expression. $$3 \sin ^{2} \theta+4 \cos ^{2} \theta=3+\cos ^{2} \theta$$

Determine whether the identity is true or false. If false, find an appropriate equivalent expression. $$\frac{\sec \theta+\tan \theta}{\cot \theta+\cos \theta}=\sec ^{2} \theta$$

Explain the basis for the cofunction identities and when they apply.

Is there only one way to evaluate \(\cos \left(\frac{5 \pi}{4}\right)\) ? Explain how to set up the solution in two different ways, and then compute to make sure they give the same answer.

Explain to someone who has forgotten the even-odd properties of sinusoidal functions how the addition and subtraction formulas can determine this characteristic for \(f(x)=\sin (x)\) and \(g(x)=\cos (x) .\) (Hint: \(0-x=-x )\)

For the following exercises, find the exact value. $$ \cos \left(\frac{7 \pi}{12}\right) $$

For the following exercises, find the exact value. $$ \cos \left(\frac{\pi}{12}\right) $$

For the following exercises, find the exact value. $$ \sin \left(\frac{5 \pi}{12}\right) $$

For the following exercises, find the exact value. $$ \sin \left(\frac{11 \pi}{12}\right) $$

For the following exercises, find the exact value. $$ \tan \left(-\frac{\pi}{12}\right) $$

For the following exercises, find the exact value. $$ \tan \left(\frac{19 \pi}{12}\right) $$

For the following exercises, rewrite in terms of \(\sin x\) and \(\cos x .\) $$ \sin \left(x+\frac{11 \pi}{6}\right) $$

For the following exercises, rewrite in terms of \(\sin x\) and \(\cos x .\) $$ \sin \left(x-\frac{3 \pi}{4}\right) $$