Chapter 6

Perform the addition or subtraction and use the fundamental identities to simplify. $$\tan x-\frac{\sec ^{2} x}{\tan x}$$

Use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. $$y=\frac{1}{\cot x+1}+\frac{1}{\tan x+1}$$

Solve the multiple-angle equation. $$\cos \frac{x}{4}=0$$

Find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\tan u=-\frac{5}{12}, \quad 3 \pi / 2

Verify the identity. $$\sin (3 \pi-x)=\sin x$$

Perform the addition or subtraction and use the fundamental identities to simplify. $$\tan x+\frac{\cos x}{1+\sin x}$$

Use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. $$y=\frac{\cos x}{1-\tan x}+\frac{\sin x \cos x}{\sin x-\cos x}$$

Solve the multiple-angle equation. $$\sin \frac{x}{2}=0$$

Find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\csc u=-\frac{5}{3}, \quad \pi

Verify the identity. $$\tan (x+\pi)-\tan (\pi-x)=2 \tan x$$

Perform the addition or subtraction and use the fundamental identities to simplify. $$\frac{\cos x}{1+\sin x}+\frac{1+\sin x}{\cos x}$$

Use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. $$y=\frac{1}{\sin x}-\frac{\cos ^{2} x}{\sin x}$$

Solve the multiple-angle equation. $$\tan 4 x=1$$

Find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\sec u=\frac{7}{2}, \quad 0

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

Perform the addition or subtraction and use the fundamental identities to simplify. $$\frac{\tan x}{1+\sec x}+\frac{1+\sec x}{\tan x}$$

Use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. $$y=\sin t+\frac{\cot ^{2} t}{\csc t}$$

Solve the multiple-angle equation. $$\tan 2 x=-1$$

Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1-\cos 6 x}{2}}$$

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

Rewrite the expression so that it is not in fractional form. $$\frac{\sin x}{\tan x}$$

Use the properties of logarithms and trigonometric identities to verify the identity. $$\ln |\cot \theta|=\ln |\cos \theta|-\ln |\sin \theta|$$

Solve the multiple-angle equation. $$\sec 4 x=2$$

Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1+\cos 4 x}{2}}$$

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

Rewrite the expression so that it is not in fractional form. $$\frac{\csc y}{\cot y}$$

Use the properties of logarithms and trigonometric identities to verify the identity. $$\ln |\sec \theta|=-\ln |\cos \theta|$$

Solve the multiple-angle equation. $$\sin 2 x=-\frac{\sqrt{3}}{2}$$

Use the half-angle formulas to simplify the expression. $$-\sqrt{\frac{1-\cos 8 x}{1+\cos 8 x}}$$

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

Rewrite the expression so that it is not in fractional form. $$\frac{\sin ^{2} y}{1-\cos y}$$

Use the cofunction identities to evaluate the expression without using a calculator. $$\sin ^{2} 25^{\circ}+\sin ^{2} 65^{\circ}$$

Solve the multiple-angle equation. $$\cos \frac{x}{2}=\frac{\sqrt{2}}{2}$$

Use the half-angle formulas to simplify the expression. $$-\sqrt{\frac{1-\cos (x-1)}{2}}$$

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

Rewrite the expression so that it is not in fractional form. $$\frac{\tan ^{2} x}{\csc x+1}$$

Use the cofunction identities to evaluate the expression without using a calculator. $$\cos ^{2} 18^{\circ}+\cos ^{2} 72^{\circ}$$

Solve the multiple-angle equation. $$\csc \frac{x}{4}=\sqrt{2}$$

Find the solutions of the equation in the interval \([0,2 \pi)\). Use a graphing utility to verify your answers. $$\sin \frac{x}{2}+\cos x=0$$

Rewrite the expression so that it is not in fractional form. $$\frac{3}{\sec x-\tan x}$$

Use the cofunction identities to evaluate the expression without using a calculator. $$\cos ^{2} 20^{\circ}+\cos ^{2} 52^{\circ}+\cos ^{2} 38^{\circ}+\cos ^{2} 70^{\circ}$$

Find the solutions of the equation in the interval \([0,2 \pi)\). Use a graphing utility to verify your answers. $$\sin \frac{x}{2}+\cos x-1=0$$

Use the cofunction identities to evaluate the expression without using a calculator. $$\sin ^{2} 18^{\circ}+\sin ^{2} 40^{\circ}+\sin ^{2} 50^{\circ}+\sin ^{2} 72^{\circ}$$

Find the solution(s) of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\) Use a graphing utility to verify your results. $$\tan (x+\pi)+2 \sin (x+\pi)=0$$

Use a graphing utility to complete the table and graph the functions in the same viewing window. Make a conjecture about \(y_{1}\) and \(y_{2}\) $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\\\\hline y_{1} & & & & & & & \\\\\hline y_{2} & & & & & & & \\\\\hline\end{array}$$ $$y_{1}=\cos \left(\frac{\pi}{2}-x\right), \quad y_{2}=\sin x$$

Powers of trigonometric functions are rewritten to be useful in calculus. Verify the identity. $$\cot ^{6} x=\cot ^{4} x \csc ^{2} x-\cot ^{4} x$$

Find the solution(s) of the equation in the interval \([\mathbf{0}, \mathbf{2} \pi)\) Use a graphing utility to verify your results. $$2 \sin \left(x+\frac{\pi}{2}\right)+3 \tan (\pi-x)=0$$

Use a graphing utility to complete the table and graph the functions in the same viewing window. Make a conjecture about \(y_{1}\) and \(y_{2}\) $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\\\\hline y_{1} & & & & & & & \\\\\hline y_{2} & & & & & & & \\\\\hline\end{array}$$ $$y_{1}=\cos x+\sin x \tan x, \quad y_{2}=\sec x$$

Powers of trigonometric functions are rewritten to be useful in calculus. Verify the identity. $$\sec ^{4} x \tan ^{2} x=\left(\tan ^{2} x+\tan ^{4} x\right) \sec ^{2} x$$

Use the product-to-sum formulas to write the product as a sum or difference. $$10 \cos 75^{\circ} \cos 15^{\circ}$$