Chapter 7

Solve the given equation. $$2 \cos ^{2} \theta-7 \cos \theta+3=0$$

Write the given expression in terms of \(x\) and \(y\) only. $$\sin \left(\tan ^{-1} x-\tan ^{-1} y\right)$$

Write the given expression as an algebraic expression in \(x\). $$\cos \left(2 \sin ^{-1} x\right)$$

Solve the given equation. $$\sin ^{2} \theta-\sin \theta-2=0$$

Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi)\) $$\tan \theta+\cot \theta=4 \sin 2 \theta$$

Write the given expression in terms of \(x\) and \(y\) only. $$\sin \left(\sin ^{-1} x+\cos ^{-1} y\right)$$

Verify the identity. $$\csc x-\sin x=\cos x \cot x$$

Find the exact value of the given expression. $$\sin \left(2 \cos ^{-1} \frac{7}{25}\right)$$

Solve the given equation. $$\cos ^{2} \theta-\cos \theta-6=0$$

Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi)\) $$\cos 2 \theta-\cos ^{2} \theta=0$$

Verify the identity. $$(\cot x-\csc x)(\cos x+1)=-\sin x$$

Find the exact value of the expression. $$\sin \left(\cos ^{-1} \frac{1}{2}+\tan ^{-1} 1\right)$$

Find the exact value of the given expression. $$\cos \left(2 \tan ^{-1} \frac{12}{3}\right)$$

Solve the given equation. $$2 \sin ^{2} \theta+5 \sin \theta-12=0$$

Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi)\) $$2 \sin ^{2} \theta=2+\cos 2 \theta$$

Verify the identity. $$\sin ^{4} \theta-\cos ^{4} \theta=\sin ^{2} \theta-\cos ^{2} \theta$$

Find the exact value of the expression. $$\cos \left(\sin ^{-1} \frac{\sqrt{3}}{2}+\cot ^{-1} \sqrt{3}\right)$$

Find the exact value of the given expression. $$\sec \left(2 \sin ^{-1} \frac{1}{4}\right)$$

Solve the given equation. $$\sin ^{2} \theta=2 \sin \theta+3$$

Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi)\) $$\cos 2 \theta-\cos 4 \theta=0$$

Verify the identity. $$\left(1-\cos ^{2} x\right)\left(1+\cot ^{2} x\right)=1$$

Find the exact value of the expression. $$\tan \left(\sin ^{-1} \frac{3}{4}-\cos ^{-1} \frac{1}{3}\right)$$

Find the exact value of the given expression. $$\tan \left(\frac{1}{2} \cos ^{-1} \frac{2}{3}\right)$$

Solve the given equation. $$3 \tan ^{3} \theta=\tan \theta$$

Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi)\) $$\sin 3 \theta-\sin 6 \theta=0$$

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

Find the exact value of the expression. $$\sin \left(\cos ^{-1} \frac{2}{3}-\tan ^{-1} \frac{1}{2}\right)$$

Solve the given equation. $$\cos \theta(2 \sin \theta+1)=0$$

Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi)\) $$\cos \theta-\sin \theta=\sqrt{2} \sin \frac{\theta}{2}$$

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

Evaluate each expression under the given conditions. \(\cos (\theta-\phi) ; \cos \theta=\frac{3}{5}, \theta\) in Quadrant IV, \(\tan \phi=-\sqrt{3}, \phi\) in Quadrant II.

Evaluate each expression under the given conditions. \(\sin (\theta / 2) ; \tan \theta=-\frac{5}{12}, \theta\) in Quadrant IV

Solve the given equation. $$\sec \theta(2 \cos \theta-\sqrt{2})=0$$

Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi)\) $$\sin \theta-\cos \theta=\frac{1}{2}$$

Verify the identity. $$(\tan y+\cot y) \sin y \cos y=1$$

Evaluate each expression under the given conditions. \(\sin 2 \theta ; \sin \theta=\frac{1}{7}, \theta\) in Quadrant II

Solve the given equation. $$\cos \theta \sin \theta-2 \cos \theta=0$$

Solve the equation by first using a Sum-to-Product Formula. $$\sin \theta+\sin 3 \theta=0$$

Verify the identity. $$\frac{1-\cos \alpha}{\sin \alpha}=\frac{\sin \alpha}{1+\cos \alpha}$$

Evaluate each expression under the given conditions. \(\tan 2 \theta ; \cos \theta=\frac{3}{5}, \theta\) in Quadrant I

Solve the given equation. $$\tan \theta \sin \theta+\sin \theta=0$$

Solve the equation by first using a Sum-to-Product Formula. $$\cos 5 \theta-\cos 7 \theta=0$$

Verify the identity. $$\sin ^{2} \alpha+\cos ^{2} \alpha+\tan ^{2} \alpha=\sec ^{2} \alpha$$

Write the product as a sum. $$\sin 2 x \cos 3 x$$

Solve the given equation. $$3 \tan \theta \sin \theta-2 \tan \theta=0$$

Solve the equation by first using a Sum-to-Product Formula. $$\cos 4 \theta+\cos 2 \theta=\cos \theta$$

Verify the identity. $$\tan ^{2} \theta-\sin ^{2} \theta=\tan ^{2} \theta \sin ^{2} \theta$$

Write the expression in terms of sine only. $$-\sqrt{3} \sin x+\cos x$$

Write the product as a sum. $$\sin x \sin 5 x$$

Solve the given equation. $$4 \cos \theta \sin \theta+3 \cos \theta=0$$