Chapter 6

Verify each identity. \(\frac{\cot ^{2} t}{\csc t}=\csc t-\sin t\)

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Find the exact value of each expression. $$ \tan \left(\frac{4 \pi}{3}-\frac{\pi}{4}\right) $$

Find all solutions of each equation. $$ 3 \sin \theta+5=-2 \sin \theta $$

verify each identity. $$ \frac{\sin 3 x-\sin x}{\cos 3 x-\cos x}=-\cot 2 x $$

Verify each identity. $$ \sin 2 \theta=\frac{2 \tan \theta}{1+\tan ^{2} \theta} $$

Verify each identity. \(\frac{1-\cos \theta}{\sin \theta}=\csc \theta-\cot \theta\)

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Find the exact value of each expression. $$ \tan \left(\frac{5 \pi}{3}-\frac{\pi}{4}\right) $$

Find all solutions of each equation. $$ 7 \cos \theta+9=-2 \cos \theta $$

verify each identity. $$ \frac{\sin x+\sin 3 x}{\cos x+\cos 3 x}=\tan 2 x $$

Verify each identity. $$ \sin 2 \theta=\frac{2 \cot \theta}{1+\cot ^{2} \theta} $$

Verify each identity. \(\frac{1-\sin \theta}{\cos \theta}=\sec \theta-\tan \theta\)

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$ \sin 25^{\circ} \cos 5^{\circ}+\cos 25^{\circ} \sin 5^{\circ} $$

Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$ \sin 2 x=\frac{\sqrt{3}}{2} $$

verify each identity. $$ \frac{\sin 2 x+\sin 4 x}{\cos 2 x+\cos 4 x}=\tan 3 x $$

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

Verify each identity. \(\frac{\sin t}{\csc t}+\frac{\cos t}{\sec t}=1\)

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$ \sin 40^{\circ} \cos 20^{\circ}+\cos 40^{\circ} \sin 20^{\circ} $$

Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$ \cos 2 x=\frac{\sqrt{2}}{2} $$

verify each identity. $$ \frac{\cos 4 x-\cos 2 x}{\sin 2 x-\sin 4 x}=\tan 3 x $$

Verify each identity. $$ (\sin \theta-\cos \theta)^{2}=1-\sin 2 \theta $$

Verify each identity. \(\frac{\sin t}{\tan t}+\frac{\cos t}{\cot t}=\sin t+\cos t\)

Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$ \cos 4 x=-\frac{\sqrt{3}}{2} $$

verify each identity. $$ \frac{\sin x-\sin y}{\sin x+\sin y}=\tan \frac{x-y}{2} \cot \frac{x+y}{2} $$

Verify each identity. $$ \sin ^{2} x+\cos 2 x=\cos ^{2} x $$

Verify each identity. \(\tan t+\frac{\cos t}{1+\sin t}=\sec t\)

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$ \frac{\tan 10^{8}+\tan 35^{\circ}}{1-\tan 10^{\circ} \tan 35^{\circ}} $$

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$ \frac{\tan 50^{\circ}-\tan 20^{\circ}}{1+\tan 50^{\circ} \tan 20^{\circ}} $$

Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$ \sin 4 x=-\frac{\sqrt{2}}{2} $$

verify each identity. $$ \frac{\sin x+\sin y}{\sin x-\sin y}=\tan \frac{x+y}{2} \cot \frac{x-y}{2} $$

Verify each identity. $$ 1-\tan ^{2} x=\frac{\cos 2 x}{\cos ^{2} x} $$

Verify each identity. \(\cot t+\frac{\sin t}{1+\cos t}=\csc t\)

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$ \sin \frac{5 \pi}{12} \cos \frac{\pi}{4}-\cos \frac{5 \pi}{12} \sin \frac{\pi}{4} $$

Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$ \tan 3 x=\frac{\sqrt{3}}{3} $$

verify each identity. $$ \frac{\sin x+\sin y}{\cos x+\cos y}=\tan \frac{x+y}{2} $$

Verify each identity. $$ \cot x=\frac{\sin 2 x}{1-\cos 2 x} $$

Verify each identity. \(1-\frac{\sin ^{2} x}{1+\cos x}=\cos x\)

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$ \sin \frac{7 \pi}{12} \cos \frac{\pi}{12}-\cos \frac{7 \pi}{12} \sin \frac{\pi}{12} $$

Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$ \tan 3 x=\sqrt{3} $$

verify each identity. $$ \frac{\sin x-\sin y}{\cos x-\cos y}=-\cot \frac{x+y}{2} $$

Verify each identity. $$ \cot x=\frac{1+\cos 2 x}{\sin 2 x} $$

Verify each identity. \(1-\frac{\cos ^{2} x}{1+\sin x}=\sin x\)

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$ \frac{\tan \frac{\pi}{5}-\tan \frac{\pi}{30}}{1+\tan \frac{\pi}{5} \tan \frac{\pi}{30}} $$

Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$ \tan \frac{x}{2}=\sqrt{3} $$

Verify each identity. $$ \sin 2 t-\tan t=\tan t \cos 2 t $$

Verify each identity. \(\frac{\cos x}{1-\sin x}+\frac{1-\sin x}{\cos x}=2 \sec x\)

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$ \frac{\tan \frac{\pi}{5}+\tan \frac{4 \pi}{5}}{1-\tan \frac{\pi}{5} \tan \frac{4 \pi}{5}} $$

Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$ \tan \frac{x}{2}=\frac{\sqrt{3}}{3} $$

Verify each identity. $$ \sin 2 t-\cot t=-\cot t \cos 2 t $$

Verify each identity. \(\frac{\sin x}{\cos x+1}+\frac{\cos x-1}{\sin x}=0\)

the graph with the given equation is shown in \(a\left[0,2 \pi, \frac{\pi}{2}\right] b y[-2,2,1]\) viewing rectangle. a. Describe the graph using another equation. b. Verify that the two equations are equivalent. $$ y=\frac{\cos x-\cos 3 x}{\sin x+\sin 3 x} $$