Chapter 8

Verify the identity. $$ \frac{\tan v \sin v}{\tan v+\sin v}=\frac{\tan v-\sin v}{\tan v \sin v} $$

\(73-90\) Prove the identity. $$ \cos ^{2} 5 x-\sin ^{2} 5 x=\cos 10 x $$

Verify the identity. $$ \sec ^{4} x-\tan ^{4} x=\sec ^{2} x+\tan ^{2} x $$

\(73-90\) Prove the identity. $$ \sin 8 x=2 \sin 4 x \cos 4 x $$

Verify the identity. $$ \frac{\cos \theta}{1-\sin \theta}=\sec \theta+\tan \theta $$

\(73-90\) Prove the identity. $$ (\sin x+\cos x)^{2}=1+\sin 2 x $$

Verify the identity. $$ \frac{\cos \theta}{1-\sin \theta}=\frac{\sin \theta-\csc \theta}{\cos \theta-\cot \theta} $$

\(73-90\) Prove the identity. $$ \frac{2 \tan x}{1+\tan ^{2} x}=\sin 2 x $$

Verify the identity. $$ \frac{1+\tan x}{1-\tan x}=\frac{\cos x+\sin x}{\cos x-\sin x} $$

\(73-90\) Prove the identity. $$ \frac{\sin 4 x}{\sin x}=4 \cos x \cos 2 x $$

Verify the identity. $$ \frac{\cos ^{2} t+\tan ^{2} t-1}{\sin ^{2} t}=\tan ^{2} t $$

Verify the identity. $$ \frac{1}{1-\sin x}-\frac{1}{1+\sin x}=2 \sec x \tan x $$

\(73-90\) Prove the identity. $$ \frac{2(\tan x-\cot x)}{\tan ^{2} x-\cot ^{2} x}=\sin 2 x $$

Verify the identity. $$ \frac{1}{\sec x+\tan x}+\frac{1}{\sec x-\tan x}=2 \sec x $$

\(73-90\) Prove the identity. $$ \cot 2 x=\frac{1-\tan ^{2} x}{2 \tan x} $$

Verify the identity. $$ \frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=4 \tan x \sec x $$

\(73-90\) Prove the identity. $$ \tan 3 x=\frac{3 \tan x-\tan ^{3} x}{1-3 \tan ^{2} x} $$

Verify the identity. $$ (\tan x+\cot x)^{2}=\sec ^{2} x+\csc ^{2} x $$

\(73-90\) Prove the identity. $$ 4\left(\sin ^{6} x+\cos ^{6} x\right)=4-3 \sin ^{2} 2 x $$

Verify the identity. $$ \tan ^{2} x-\cot ^{2} x=\sec ^{2} x-\csc ^{2} x $$

\(73-90\) Prove the identity. $$ \cos ^{4} X-\sin ^{4} X=\cos 2 X $$

Verify the identity. $$ \frac{\sec u-1}{\sec u+1}=\frac{1-\cos u}{1+\cos u} $$

\(73-90\) Prove the identity. $$ \tan ^{2}\left(\frac{x}{2}+\frac{\pi}{4}\right)=\frac{1+\sin x}{1-\sin x} $$

Verify the identity. $$ \frac{\cot x+1}{\cot x-1}=\frac{1+\tan x}{1-\tan x} $$

\(73-90\) Prove the identity. $$ \frac{\sin x+\sin 5 x}{\cos x+\cos 5 x}=\tan 3 x $$

Verify the identity. $$ \frac{\sin ^{3} x+\cos ^{3} x}{\sin x+\cos x}=1-\sin x \cos x $$

\(73-90\) Prove the identity. $$ \frac{\sin 3 x+\sin 7 x}{\cos 3 x-\cos 7 x}=\cot 2 x $$

Verify the identity. $$ \frac{\tan v-\cot v}{\tan ^{2} v-\cot ^{2} v}=\sin v \cos v $$

\(73-90\) Prove the identity. $$ \frac{\sin 10 x}{\sin 9 x+\sin x}=\frac{\cos 5 x}{\cos 4 x} $$

Verify the identity. $$ \frac{1+\sin X}{1-\sin X}=(\tan X+\sec X)^{2} $$

\(73-90\) Prove the identity. $$ \frac{\sin x+\sin 3 x+\sin 5 x}{\cos x+\cos 3 x+\cos 5 x}=\tan 3 x $$

Verify the identity. $$ \frac{\tan x+\tan y}{\cot x+\cot y}=\tan x \tan y $$

\(73-90\) Prove the identity. $$ \frac{\sin x+\sin y}{\cos x+\cos y}=\tan \left(\frac{x+y}{2}\right) $$

Verify the identity. $$ (\tan x+\cot x)^{4}=\csc ^{4} x \sec ^{4} x $$

\(73-90\) Prove the identity. $$ \tan y=\frac{\sin (x+y)-\sin (x-y)}{\cos (x+y)+\cos (x-y)} $$

Verify the identity. $$ (\sin \alpha-\tan \alpha)(\cos \alpha-\cot \alpha)=(\cos \alpha-1)(\sin \alpha-1) $$

Show that \(\sin 130^{\circ}-\sin 110^{\circ}=-\sin 10^{\circ}\).

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7\()\) . Assume that \(0 \leq \theta<\pi / 2 .\) $$ \frac{x}{\sqrt{1-x^{2}}}, \quad x=\sin \theta $$

Show that \(\cos 100^{\circ}-\cos 200^{\circ}=\sin 50^{\circ}\).

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7\()\) . Assume that \(0 \leq \theta<\pi / 2 .\) $$ \sqrt{1+x^{2}}, \quad x=\tan \theta $$

Show that \(\sin 45^{\circ}+\sin 15^{\circ}=\sin 75^{\circ}\).

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7\()\) . Assume that \(0 \leq \theta<\pi / 2 .\) $$ \sqrt{x^{2}-1}, \quad x=\sec \theta $$

Show that \(\cos 87^{\circ}+\cos 33^{\circ}=\sin 63^{\circ}\).

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7\()\) . Assume that \(0 \leq \theta<\pi / 2 .\) $$ \frac{1}{x^{2} \sqrt{4+x^{2}}}, \quad x=2 \tan \theta $$

Prove the identity $$ \frac{\sin x+\sin 2 x+\sin 3 x+\sin 4 x+\sin 5 x}{\cos x+\cos 2 x+\cos 3 x+\cos 4 x+\cos 5 x}=\tan 3 x $$

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7\()\) . Assume that \(0 \leq \theta<\pi / 2 .\) $$ \sqrt{9-x^{2}}, \quad x=3 \sin \theta $$

Use the identity $$\sin 2 x=2 \sin x \cos x$$ \(n\) times to show that $$\sin \left(2^{n} x\right)=2^{n} \sin x \cos x \cos 2 x \cos 4 x \cdot \cdot \cos 2^{n-1} x$$

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7\()\) . Assume that \(0 \leq \theta<\pi / 2 .\) $$ \frac{\sqrt{x^{2}-25}}{x}, \quad x=5 \sec \theta $$

Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x)=g(x)\) is an identity? Prove your answer. $$ f(x)=\cos ^{2} x-\sin ^{2} x, \quad g(x)=1-2 \sin ^{2} x $$

(a) Graph \(f(x)=\cos 2 x+2 \sin ^{2} x\) and make a conjecture. (b) Prove the conjecture you made in part (a).