Chapter 8

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

\(43-52\) a Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi) .\) \(\cos 2 \theta-\cos ^{2} \theta=0\)

\(47-50\) Find the exact value of the given expression. $$ \sin \left(2 \cos ^{-1} \frac{7}{25}\right) $$

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

\(39-56 \approx\) Solve the given equation. $$ \cos ^{2} \theta-\cos \theta-6=0 $$

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

\(43-52\) a Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi) .\) \(2 \sin ^{2} \theta=2+\cos 2 \theta\)

\(47-50\) Find the exact value of the given expression. $$ \cos \left(2 \tan ^{-1} \frac{12}{5}\right) $$

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

\(39-56 \approx\) Solve the given equation. $$ 2 \sin ^{2} \theta+5 \sin \theta-12=0 $$

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

\(43-52\) a Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi) .\) \(\cos 2 \theta-\cos 4 \theta=0\)

\(47-50\) Find the exact value of the given expression. $$ \sec \left(2 \sin ^{-1} \frac{1}{4}\right) $$

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

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

\(43-52\) a Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi) .\) \(\sin 3 \theta-\sin 6 \theta=0\)

\(47-50\) Find the exact value of the given expression. $$ \tan \left(\frac{1}{2} \cos ^{-1} \frac{2}{3}\right) $$

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

\(39-56 \approx\) Solve the given equation. $$ 3 \tan ^{3} \theta=\tan \theta $$

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

\(43-52\) a Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi) .\) \(\cos \theta-\sin \theta=\sqrt{2} \sin \frac{\theta}{2}\)

\(51-54\). Evaluate each expression under the given conditions. $$ \cos 2 \theta ; \sin \theta=-\frac{3}{5}, \theta \text { in Quadrant III } $$

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.

\(39-56 \approx\) Solve the given equation. $$ \cos \theta(2 \sin \theta+1)=0 $$

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

\(43-52\) a Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi) .\) \(\sin \theta-\cos \theta=\frac{1}{2}\)

\(51-54\). Evaluate each expression under the given conditions. $$ \sin (\theta / 2) ; \tan \theta=-\frac{5}{12}, \theta \text { in Quadrant IV } $$

Evaluate each expression under the given conditions. \(\sin (\theta-\phi) ; \tan \theta=\frac{4}{3}, \theta\) in Quadrant III, \(\sin \phi=-\sqrt{10} / 10\) \(\phi\) in Quadrant IV

\(39-56 \approx\) Solve the given equation. $$ \sec \theta(2 \cos \theta-\sqrt{2})=0 $$

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

\(53-56\) a Solve the equation by first using a Sum-to-Product Formula. \(\sin \theta+\sin 3 \theta=0\)

\(51-54\). Evaluate each expression under the given conditions. $$ \sin 2 \theta ; \sin \theta=\frac{1}{7}, \theta \text { in Quadrant II } $$

Evaluate each expression under the given conditions. \(\sin (\theta+\phi) ; \sin \theta=\frac{5}{13}, \theta\) in Quadrant \(\mathrm{I}, \cos \phi=-2 \sqrt{5} / 5, \phi\) in Quadrant II

\(39-56 \approx\) Solve the given equation. $$ \cos \theta \sin \theta-2 \cos \theta=0 $$

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

\(53-56\) a Solve the equation by first using a Sum-to-Product Formula. \(\cos 5 \theta-\cos 7 \theta=0\)

\(51-54\). Evaluate each expression under the given conditions. $$ \tan 2 \theta ; \cos \theta=\frac{3}{5}, \theta \text { in Quadrant I } $$

Evaluate each expression under the given conditions. \(\tan (\theta+\phi) ; \cos \theta=-\frac{1}{3}, \theta\) in Quadrant III, \(\sin \phi=\frac{1}{4}, \phi\) in Quadrant II

\(39-56 \approx\) Solve the given equation. $$ \tan \theta \sin \theta+\sin \theta=0 $$

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

\(53-56\) a Solve the equation by first using a Sum-to-Product Formula. \(\cos 4 \theta+\cos 2 \theta=\cos \theta\)

\(55-60\) Write the product as a sum. $$ \sin 2 x \cos 3 x $$

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

\(39-56 \approx\) Solve the given equation. $$ 3 \tan \theta \sin \theta-2 \tan \theta=0 $$

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

\(53-56\) a Solve the equation by first using a Sum-to-Product Formula. \(\sin 5 \theta-\sin 3 \theta=\cos 4 \theta\)

\(55-60\) Write the product as a sum. $$ \sin x \sin 5 x $$

Write the expression in terms of sine only. $$ \sin x+\cos x $$

\(39-56 \approx\) Solve the given equation. $$ 4 \cos \theta \sin \theta+3 \cos \theta=0 $$

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