Chapter 10

Verify that each equation is an identity. $$\tan x=\csc 2 x-\cot 2 x$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$3 \sin ^{2} \theta-\sin \theta=2$$

Perform indicated operation and simplify the result. $$(1+\tan s)^{2}-2 \tan s$$

Give the exact real number value of each expression. Do not use a calculator. $$\cos ^{-1}\left(\cos \frac{3 \pi}{2}\right)$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$2 \cos ^{2} x+\cos x-1=0$$

Perform indicated operation and simplify the result. $$\frac{1}{1+\cos x}-\frac{1}{1-\cos x}$$

Give the exact real number value of each expression. Do not use a calculator. $$\sin ^{-1}\left(\sin \frac{3 \pi}{2}\right)$$

Verify that each equation is an identity. $$\sin 4 \gamma=4 \sin \gamma \cos \gamma-8 \sin ^{3} \gamma \cos \gamma$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$4 \cos ^{2} x-1=0$$

Perform indicated operation and simplify the result. $$(\sin \alpha-\cos \alpha)^{2}$$

Give the exact real number value of each expression. Do not use a calculator. $$\tan \left(\tan ^{-1} 5\right)$$

Verify that each equation is an identity. $$\sin 4 \alpha=4 \sin \alpha \cos \alpha \cos 2 \alpha$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\sin \theta \cos \theta-\sin \theta=0$$

Verify that equation is an identity. \(\frac{\cot \theta}{\csc \theta}=\cos \theta\)

Give the exact real number value of each expression. Do not use a calculator. $$\tan \left(\tan ^{-1}(-4)\right)$$

Verify that each equation is an identity. $$\frac{\sin 2 x}{2 \sin x}=\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\tan \theta \csc \theta-\sqrt{3} \csc \theta=0$$

Verify that equation is an identity. \(\frac{\tan \theta}{\sec \theta}=\sin \theta\)

Give the exact real number value of each expression. Do not use a calculator. $$\sec \left(\sec ^{-1} 2\right)$$

Write each expression as a sum or difference of trigonometric functions or values. $$2 \sin 58^{\circ} \cos 102^{\circ}$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\sin x(3 \sin x-1)=1$$

Verify that equation is an identity. \(\cos ^{2} \theta\left(\tan ^{2} \theta+1\right)=1\)

Give the exact real number value of each expression. Do not use a calculator. $$\csc \left(\csc ^{-1} 3\right)$$

Write each expression as a sum or difference of trigonometric functions or values. $$5 \cos 3 x \cos 2 x$$

Verify that equation is an identity. \(\frac{\cos \theta+1}{\tan ^{2} \theta}=\frac{\cos \theta}{\sec \theta-1}\)

Give the exact real number value of each expression. Do not use a calculator. $$\tan ^{-1}\left(\tan \frac{5 \pi}{6}\right)$$

Write each expression as a sum or difference of trigonometric functions or values. $$2 \cos 85^{\circ} \sin 140^{\circ}$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$5+5 \tan ^{2} \theta=6 \sec \theta$$

Verify that equation is an identity. \(\frac{\tan ^{2} \gamma+1}{\sec \gamma}=\sec \gamma\)

Give the exact real number value of each expression. Do not use a calculator. $$\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$$

Write each expression as a sum or difference of trigonometric functions or values. $$\sin 4 x \sin 5 x$$

Verify that equation is an identity. \(\sin ^{2} \beta\left(1+\cot ^{2} \beta\right)=1\)

Give the exact real number value of each expression. Do not use a calculator. $$\tan \left(\arccos \frac{3}{4}\right)$$

Write each expression as a product of trigonometric functions or values. $$\cos 4 x-\cos 2 x$$

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\frac{2 \tan \theta}{3-\tan ^{2} \theta}=1$$

Verify that equation is an identity. \(\frac{1-\sin ^{2} \beta}{\cos \beta}=\cos \beta\)

Give the exact real number value of each expression. Do not use a calculator. $$\sin \left(\arccos \frac{1}{4}\right)$$

Write each expression as a product of trigonometric functions or values. $$\cos 5 t+\cos 8 t$$

How many solutions does the equation \(2 x-1=0\) have? How many solutions does the equation \(2 \sin x-1=0\) have? Explain.

Verify that equation is an identity. \(\frac{\sin ^{2} \theta}{\cos \theta}=\sec \theta-\cos \theta\)

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(\tan ^{-1}(-2)\right)$$

Write each expression as a product of trigonometric functions or values. $$\sin 25^{\circ}+\sin \left(-48^{\circ}\right)$$

Express solutions to the nearest hundredth. (Hint: In Exercise 83 , the equation has three solutions.) $$2 \sin x=1-2 \cos x$$

Verify that equation is an identity. \(\frac{1-\cos x}{1+\cos x}=(\cot x-\csc x)^{2}\)

Give the exact real number value of each expression. Do not use a calculator. $$\sec \left(\sin ^{-1}\left(-\frac{1}{5}\right)\right)$$

Write each expression as a product of trigonometric functions or values. $$\sin 102^{\circ}-\sin 95^{\circ}$$

Express solutions to the nearest hundredth. (Hint: In Exercise 83 , the equation has three solutions.) $$\cot x+2 \csc x=3$$

Verify that equation is an identity. \(\frac{\cot ^{2} t-1}{1+\cot ^{2} t}=1-2 \sin ^{2} t\)

Give the exact real number value of each expression. Do not use a calculator. $$\sin \left(2 \tan ^{-1} \frac{12}{5}\right)$$

Write each expression as a product of trigonometric functions or values. $$\cos 4 x+\cos 8 x$$