Chapter 6

Rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\cos ^{8} x$$

Find the exact value of the expression. $$\cos 120^{\circ} \cos 30^{\circ}+\sin 120^{\circ} \sin 30^{\circ}$$

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\frac{\sec \theta}{\csc \theta}$$

Verify the identity. $$(1+\sin y)[1+\sin (-y)]=\cos ^{2} y$$

Find all solutions of the equation in the interval \([0,2 \pi)\) algebraically. Use the table feature of a graphing utility to check your answers numerically. $$3 \sin x+1=\sin x$$

Rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\sin ^{2} x \cos ^{4} x$$

Find the exact value of the expression. $$\frac{\tan (5 \pi / 6)-\tan (\pi / 6)}{1+\tan (5 \pi / 6) \tan (\pi / 6)}$$

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\sec \alpha \cdot \frac{\sin \alpha}{\tan \alpha}$$

Verify the identity. $$\sin ^{1 / 2} x \cos x-\sin ^{5 / 2} x \cos x=\cos ^{3} x \sqrt{\sin x}$$

Find all solutions of the equation in the interval \([0,2 \pi)\) algebraically. Use the table feature of a graphing utility to check your answers numerically. $$\csc ^{2} x-2=0$$

Rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\sin ^{4} x \cos ^{2} x$$

Find the exact value of the expression. $$\frac{\tan 25^{\circ}+\tan 110^{\circ}}{1-\tan 25^{\circ} \tan 110^{\circ}}$$

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\frac{1+\tan ^{2} \theta}{\sec ^{2} \theta}$$

Verify the identity. $$\sec ^{6} x(\sec x \tan x)-\sec ^{4} x(\sec x \tan x)=\sec ^{5} x \tan ^{3} x$$

Find all solutions of the equation in the interval \([0,2 \pi)\) algebraically. Use the table feature of a graphing utility to check your answers numerically. $$\tan ^{2} x-1=0$$

Rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\sin ^{2} 2 x$$

Use a graphing utility to complete the table and graph the two functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2} .\) Then verify the identity algebraically. $$y_{1}=\cos (x+\pi) \cos (x-\pi), \quad y_{2}=\cos ^{2} x$$

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\sin \left(\frac{\pi}{2}-x\right) \csc x$$

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. $$\frac{\cos x-\cos y}{\sin x+\sin y}+\frac{\sin x-\sin y}{\cos x+\cos y}=0$$

Find all solutions of the equation in the interval \([0,2 \pi)\) algebraically. Use the table feature of a graphing utility to check your answers numerically. $$\cos ^{3} x=\cos x$$

Rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\cos ^{2} 2 x$$

Use a graphing utility to complete the table and graph the two functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2} .\) Then verify the identity algebraically. $$y_{1}=\sin (x+\pi) \sin (x-\pi), \quad y_{2}=\sin ^{2} x$$

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\cot \left(\frac{\pi}{2}-x\right) \cos x$$

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. $$\frac{\tan x+\cot y}{\tan x \cot y}=\tan y+\cot x$$

Rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\cos ^{2} \frac{x}{2}$$

Find the exact value of the trigonometric expression when \(\sin u=\frac{5}{13}\) and \(\cos v=-\frac{3}{5} \cdot(\text { Both } u\) and \(v\) are in Quadrant II.) $$\sin (u+v)$$

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\frac{\cos ^{2} y}{1-\sin y}$$

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. $$\frac{\cos \theta}{1-\sin \theta}=\sec \theta+\tan \theta$$

Find the exact value of the trigonometric expression when \(\sin u=\frac{5}{13}\) and \(\cos v=-\frac{3}{5} \cdot(\text { Both } u\) and \(v\) are in Quadrant II.) $$\tan (u+v)$$

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\frac{1}{\cot ^{2} x+1}$$

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. $$(\sec \theta-\tan \theta)(\csc \theta+1)=\cot \theta$$

Find the exact value of the trigonometric expression when \(\sin u=\frac{5}{13}\) and \(\cos v=-\frac{3}{5} \cdot(\text { Both } u\) and \(v\) are in Quadrant II.) $$\cos (u-v)$$

Use a graphing utility to check your result graphically. $$\cot ^{2} x-\cot ^{2} x \cos ^{2} x$$

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. $$\sin ^{2}\left(\frac{\pi}{2}-x\right)+\sin ^{2} x=1$$

Find all solutions of the equation in the interval \([0,2 \pi)\) algebraically. Use the table feature of a graphing utility to check your answers numerically. $$2 \sin x+\csc x=0$$

Find the exact value of the trigonometric expression when \(\sin u=\frac{5}{13}\) and \(\cos v=-\frac{3}{5} \cdot(\text { Both } u\) and \(v\) are in Quadrant II.) $$\sin (u-v)$$

Use a graphing utility to check your result graphically. $$\sec ^{2} x \tan ^{2} x+\sec ^{2} x$$

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. $$\sec ^{2} y-\cot ^{2}\left(\frac{\pi}{2}-y\right)=1$$

Find all solutions of the equation in the interval \([0,2 \pi)\) algebraically. Use the table feature of a graphing utility to check your answers numerically. $$\sec x+\tan x=1$$

Find the exact value of the trigonometric expression when \(\sin u=-\frac{7}{25}\) and \(\cos v=-\frac{4}{5} .\) (Both \(u\) and \(v\) are in Quadrant III.) $$\tan (u+v)$$

Use a graphing utility to check your result graphically. $$\frac{\cos ^{2} x-4}{\cos x-2}$$

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. $$\sin x \csc x\left(\frac{\pi}{2}-x\right)=\tan x$$

Rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\cos ^{4} \frac{x}{2}$$

Find the exact value of the trigonometric expression when \(\sin u=-\frac{7}{25}\) and \(\cos v=-\frac{4}{5} .\) (Both \(u\) and \(v\) are in Quadrant III.) $$\cos (u+v)$$

Use a graphing utility to check your result graphically. $$\frac{\csc ^{2} x-1}{\csc x-1}$$

Verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. $$\sec ^{2}\left(\frac{\pi}{2}-x\right)-1=\cot ^{2} x$$

Find the exact value of the trigonometric expression when \(\sin u=-\frac{7}{25}\) and \(\cos v=-\frac{4}{5} .\) (Both \(u\) and \(v\) are in Quadrant III.) $$\sin (v-u)$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$2 \sec ^{2} x-2 \sec ^{2} x \sin ^{2} x-\sin ^{2} x-\cos ^{2} x=1$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\) by collecting all terms on one side, graphing the new equation, and using the zero or root feature to approximate the \(x\) -intercepts of the graph. $$2 \sin ^{2} x+3 \sin x=-1$$

Find the exact value of the trigonometric expression when \(\sin u=-\frac{7}{25}\) and \(\cos v=-\frac{4}{5} .\) (Both \(u\) and \(v\) are in Quadrant III.) $$\cos (v-u)$$