Chapter 7

Exer. 1-50: Verify the identity. $$ \frac{\cot x}{\csc x+1}=\frac{\csc x-1}{\cot x} $$

Verify the identity. $$ \frac{\sin \theta+\sin 3 \theta}{\cos \theta+\cos 3 \theta}=\tan 2 \theta $$

If \(\cos \alpha=\frac{24}{25}\) and \(\sin \alpha<0\), find the exact value of \(\cos \left(\alpha+\frac{\pi}{6}\right)\).

Verify the identity. $$ \sin 4 t=4 \sin t \cos t\left(1-2 \sin ^{2} t\right) $$

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left[2 \arccos \left(-\frac{3}{5}\right)\right]\) (b) \(\cos \left(2 \sin ^{-1} \frac{15}{17}\right)\) (c) \(\tan \left(2 \tan ^{-1} \frac{3}{4}\right)\)

Exer. 1-38: Find all solutions of the equation. $$ 2 \cos t+1=0 $$

Exer. 1-50: Verify the identity. $$ \frac{\cot 4 u-1}{\cot 4 u+1}=\frac{1-\tan 4 u}{1+\tan 4 u} $$

If \(\alpha\) and \(\beta\) are acute angles such that \(\cos \alpha=\frac{4}{5}\) and \(\tan \beta=\frac{8}{15}\), find (a) \(\sin (\alpha+\beta)\) (b) \(\cos (\alpha+\beta)\) (c) the quadrant containing \(\alpha+\beta\)

Verify the identity. $$ \cos 4 \theta=8 \cos ^{4} \theta-8 \cos ^{2} \theta+1 $$

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left(2 \tan ^{-1} \frac{5}{12}\right)\) (b) \(\cos \left(2 \arccos \frac{9}{41}\right)\) (c) \(\tan \left[2 \arcsin \left(-\frac{8}{17}\right)\right]\)

Exer. 1-38: Find all solutions of the equation. $$ \cot \theta+1=0 $$

Exer. 1-50: Verify the identity. $$ \frac{1+\sec 4 x}{\sin 4 x+\tan 4 x}=\csc 4 x $$

Verify the identity. $$ \frac{\sin u-\sin v}{\cos u-\cos v}=-\cot \frac{1}{2}(u+v) $$

If \(\alpha\) and \(\beta\) are acute angles such that \(\csc \alpha=\frac{13}{12}\) and \(\cot \beta=\frac{4}{3}\), find (a) \(\sin (\alpha+\beta)\) (b) \(\tan (\alpha+\beta)\) (c) the quadrant containing \(\alpha+\beta\)

Verify the identity. $$ \cos 6 t=32 \cos ^{6} t-48 \cos ^{4} t+18 \cos ^{2} t-1 $$

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left[\frac{1}{2} \sin ^{-1}\left(-\frac{7}{25}\right)\right]\) (b) \(\cos \left(\frac{1}{2} \tan ^{-1} \frac{8}{15}\right)\) (c) \(\tan \left(\frac{1}{2} \cos ^{-1} \frac{3}{5}\right)\)

Exer. 1-38: Find all solutions of the equation. $$ \tan ^{2} x=1 $$

Exer. 1-50: Verify the identity. $$ \sin ^{4} r-\cos ^{4} r=\sin ^{2} r-\cos ^{2} r $$

Verify the identity. $$ \frac{\sin u-\sin v}{\sin u+\sin v}=\frac{\tan \frac{1}{2}(u-v)}{\tan \frac{1}{2}(u+v)} $$

If \(\sin \alpha=-\frac{4}{5}\) and \(\sec \beta=\frac{5}{3}\) for a third- quadrant angle \(\alpha\) and a first-quadrant angle \(\beta\), find (a) \(\sin (\alpha+\beta)\) (b) \(\tan (\alpha+\beta)\) (c) the quadrant containing \(\alpha+\beta\)

Verify the identity. $$ \sin ^{4} t=\frac{3}{8}-\frac{1}{2} \cos 2 t+\frac{1}{8} \cos 4 t $$

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left[\frac{1}{2} \cos ^{-1}\left(-\frac{3}{5}\right)\right]\) (b) \(\cos \left(\frac{1}{2} \sin ^{-1} \frac{12}{13}\right)\) (c) \(\tan \left(\frac{1}{2} \tan ^{-1} \frac{40}{9}\right)\)

Exer. 1-38: Find all solutions of the equation. $$ 4 \cos \theta-2=0 $$

Exer. 1-50: Verify the identity. $$ \sin ^{4} \theta+2 \sin ^{2} \theta \cos ^{2} \theta+\cos ^{4} \theta=1 $$

If \(\tan \alpha=-\frac{7}{24}\) and \(\cot \beta=\frac{3}{4}\) for a second- quadrant angle \(\alpha\) and a third-quadrant angle \(\beta\), find (a) \(\sin (\alpha+\beta)\) (b) \(\cos (\alpha+\beta)\) (c) \(\tan (\alpha+\beta)\) (d) \(\sin (\alpha-\beta)\) (e) \(\cos (\alpha-\beta)\) (f) \(\tan (\alpha-\beta)\)

Verify the identity. $$ \frac{\cos u-\cos v}{\cos u+\cos v}=-\tan \frac{1}{2}(u+v) \tan \frac{1}{2}(u-v) $$

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

Exer. 23-30: Write the expression as an algebraic expression in \(x\) for \(x>0\). $$ \sin \left(\tan ^{-1} x\right) $$

Exer. 1-38: Find all solutions of the equation. $$ (\cos \theta-1)(\sin \theta+1)=0 $$

Exer. 1-50: Verify the identity. $$ \tan ^{4} k-\sec ^{4} k=1-2 \sec ^{2} k $$

Verify the identity. $$ 4 \cos x \cos 2 x \sin 3 x=\sin 2 x+\sin 4 x+\sin 6 x $$

Verify the identity. $$ \sec 2 \theta=\frac{\sec ^{2} \theta}{2-\sec ^{2} \theta} $$

Exer. 23-30: Write the expression as an algebraic expression in \(x\) for \(x>0\). $$ \tan (\arccos x) $$

Exer. 1-38: Find all solutions of the equation. $$ 2 \cos x=\sqrt{3} $$

Exer. 1-50: Verify the identity. $$ \sec ^{4} u-\sec ^{2} u=\tan ^{2} u+\tan ^{4} u $$

If \(\alpha\) and \(\beta\) are second-quadrant angles such that \(\sin \alpha=\frac{2}{3}\) and \(\cos \beta=-\frac{1}{3}\), find (a) \(\sin (\alpha+\beta)\) (b) \(\tan (\alpha+\beta)\) (c) the quadrant containing \(\alpha+\beta\)

Verify the identity. $$ \frac{\cos t+\cos 4 t+\cos 7 t}{\sin t+\sin 4 t+\sin 7 t}=\cot 4 t $$

Verify the identity. $$ \cot 2 u=\frac{\cot ^{2} u-1}{2 \cot u} $$

Exer. 23-30: Write the expression as an algebraic expression in \(x\) for \(x>0\). $$ \left(\sin ^{-1} \frac{x}{\sqrt{x^{2}+4}}\right) $$

Exer. 1-38: Find all solutions of the equation. $$ \sec ^{2} \alpha-4=0 $$

Exer. 1-50: Verify the identity. $$ (\sec t+\tan t)^{2}=\frac{1+\sin t}{1-\sin t} $$

Exer. 25-36: Verify the reduction formula. $$ \sin (\theta+\pi)=-\sin \theta $$

Express as a sum. $$ (\sin a x)(\cos b x) $$

Verify the identity. $$ 2 \sin ^{2} 2 t+\cos 4 t=1 $$

Exer. 23-30: Write the expression as an algebraic expression in \(x\) for \(x>0\). $$ \cot \left(\sin ^{-1} \frac{\sqrt{x^{2}-9}}{x}\right) $$

Exer. 1-38: Find all solutions of the equation. $$ 3-\tan ^{2} \beta=0 $$

Exer. 25-36: Verify the reduction formula. $$ \sin \left(x+\frac{\pi}{2}\right)=\cos x $$

Express as a sum. $$ (\cos a u)(\cos b u) $$

Verify the identity. $$ \tan \theta+\cot \theta=2 \csc 2 \theta $$

Exer. 23-30: Write the expression as an algebraic expression in \(x\) for \(x>0\). $$ \sin \left(2 \sin ^{-1} x\right) $$