Chapter 8

Prove the identity. $$ \cos \left(x-\frac{\pi}{2}\right)=\sin x $$

\(25-38\) . Find all solutions of the given equation. $$ \sin \theta+1=0 $$

\(17-28\) Use an appropriate Half-Angle Formula to find the exact value of the expression. $$ \tan \frac{5 \pi}{12} $$

Simplify the trigonometric expression. $$ \frac{\cos x}{\sec x+\tan x} $$

\(17-34\) . An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi) .\) $$ \sin 2 \theta=3 \cos 2 \theta $$

Prove the identity. $$ \sin (x-\pi)=-\sin x $$

\(25-38\) . Find all solutions of the given equation. $$ \sqrt{2} \sin \theta+1=0 $$

\(17-28\) Use an appropriate Half-Angle Formula to find the exact value of the expression. $$ \sin \frac{9 \pi}{8} $$

Consider the given equation. (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity. $$ \frac{\cos x}{\sec x \sin x}=\csc x-\sin x $$

\(17-34\) . An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi) .\) $$ \csc 3 \theta=5 \sin 3 \theta $$

Prove the identity. $$ \cos (x-\pi)=-\cos x $$

\(25-38\) . Find all solutions of the given equation. $$ \sqrt{2} \cos \theta-1=0 $$

\(17-28\) Use an appropriate Half-Angle Formula to find the exact value of the expression. $$ \sin \frac{11 \pi}{12} $$

Consider the given equation. (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity. $$ \frac{\tan y}{\csc y}=\sec y-\cos y $$

\(17-34\) . An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi) .\) $$ \sec \theta-\tan \theta=\cos \theta $$

Prove the identity. $$ \cos (x-\pi)=-\cos x $$

\(25-38\) . Find all solutions of the given equation. $$ 5 \sin \theta-1=0 $$

\(29-34\) Simplify the expression by using a Double-Angle Formula or a Half- Angle Formula. $$ \begin{array}{ll}{\text { (a) } 2 \sin 18^{\circ} \cos 18^{\circ}} & {\text { (b) } 2 \sin 3 \theta \cos 3 \theta}\end{array} $$

Verify the identity. $$ \frac{\sin \theta}{\tan \theta}=\cos \theta $$

\(17-34\) . An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi) .\) $$ \tan 3 \theta+1=\sec 3 \theta $$

\(29-34\) Simplify the expression by using a Double-Angle Formula or a Half- Angle Formula. $$ \begin{array}{ll}{\text { (a) } \frac{2 \tan 7^{\circ}}{1-\tan ^{2} 7^{\circ}}} & {\text { (b) } \frac{2 \tan 7 \theta}{1-\tan ^{2} 7 \theta}}\end{array} $$

\(25-38\) . Find all solutions of the given equation. $$ 4 \cos \theta+1=0 $$

Verify the identity. $$ \frac{\tan x}{\sec x}=\sin x $$

\(17-34\) . An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi) .\) $$ 3 \tan ^{3} \theta-3 \tan ^{2} \theta-\tan \theta+1=0 $$

Prove the identity. $$ \cos \left(x+\frac{\pi}{6}\right)+\sin \left(x-\frac{\pi}{3}\right)=0 $$

\(29-34\) Simplify the expression by using a Double-Angle Formula or a Half- Angle Formula. $$ \text { (a) } \cos ^{2} 34^{\circ}-\sin ^{2} 34^{\circ} \quad \text { (b) } \cos ^{2} 5 \theta-\sin ^{2} 5 \theta $$

\(25-38\) . Find all solutions of the given equation. $$ 3 \tan ^{2} \theta-1=0 $$

Verify the identity. $$ \frac{\cos u \sec u}{\tan u}=\cot u $$

\(17-34\) . An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi) .\) $$ 4 \sin \theta \cos \theta+2 \sin \theta-2 \cos \theta-1=0 $$

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

\(29-34\) Simplify the expression by using a Double-Angle Formula or a Half- Angle Formula. $$ \begin{array}{ll}{\text { (a) } \cos ^{2} \frac{\theta}{2}-\sin ^{2} \frac{\theta}{2}} & {\text { (b) } 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}\end{array} $$

\(25-38\) . Find all solutions of the given equation. $$ \cot \theta+1=0 $$

Verify the identity. $$ \frac{\cot x \sec x}{\csc x}=1 $$

\(17-34\) . An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi) .\) $$ 2 \sin \theta \tan \theta-\tan \theta=1-2 \sin \theta $$

Prove the identity. $$ \sin (x+y)-\sin (x-y)=2 \cos x \sin y $$

\(29-34\) Simplify the expression by using a Double-Angle Formula or a Half- Angle Formula. $$ \begin{array}{ll}{\text { (a) } \frac{\sin 8^{\circ}}{1+\cos 8^{\circ}}} & {\text { (b) } \frac{1-\cos 4 \theta}{\sin 4 \theta}}\end{array} $$

\(25-38\) . Find all solutions of the given equation. $$ 2 \cos ^{2} \theta-1=0 $$

Verify the identity. $$ \sin B+\cos B \cot B=\csc B $$

\(17-34\) . An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0,2 \pi) .\) $$ \sec \theta \tan \theta-\cos \theta \cot \theta=\sin \theta $$

Prove the identity. $$ \cos (x+y)+\cos (x-y)=2 \cos x \cos y $$

\(29-34\) Simplify the expression by using a Double-Angle Formula or a Half- Angle Formula. $$ \text { (a) } \sqrt{\frac{1-\cos 30^{\circ}}{2}} \quad \text { (b) } \sqrt{\frac{1-\cos 8 \theta}{2}} $$

\(25-38\) . Find all solutions of the given equation. $$ 4 \sin ^{2} \theta-3=0 $$

Verify the identity. $$ \cos (-x)-\sin (-x)=\cos x+\sin x $$

\(35-38=(a)\) Graph \(f\) and \(g\) in the given viewing rectangle and find the intersection points graphically, rounded to two decimal places. (b) Find the intersection points of \(f\) and \(g\) algebraically. Give exact answers. $$ \begin{array}{l}{f(x)=3 \cos x+1, g(x)=\cos x-1} \\ {[-2 \pi, 2 \pi] \text { by }[-2.5,4.5]}\end{array} $$

Prove the identity. $$ \cot (x-y)=\frac{\cot x \cot y+1}{\cot y-\cot x} $$

Use the Addition Formula for Sine to prove the Double-Angle Formula for Sine.

\(25-38\) . Find all solutions of the given equation. $$ \tan ^{2} \theta-4=0 $$

Verify the identity. $$ \cot (-\alpha) \cos (-\alpha)+\sin (-\alpha)=-\csc \alpha $$

\(35-38=(a)\) Graph \(f\) and \(g\) in the given viewing rectangle and find the intersection points graphically, rounded to two decimal places. (b) Find the intersection points of \(f\) and \(g\) algebraically. Give exact answers. $$ \begin{array}{l}{f(x)=\sin 2 x+1, g(x)=2 \sin 2 x+1} \\ {[-2 \pi, 2 \pi] \text { by }[-1.5,3.5]}\end{array} $$

Prove the identity. $$ \cot (x+y)=\frac{\cot x \cot y-1}{\cot x+\cot y} $$