Chapter 7

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

Exer. 1-50: Verify the identity. $$ \left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{3}=1 $$

Exer. 25-36: Verify the reduction formula. $$ \sin \left(x-\frac{5 \pi}{2}\right)=-\cos x $$

Use sum-to-product formulas to find the solutions of the equation. $$ \sin 5 t+\sin 3 t=0 $$

Verify the identity. $$ \tan 3 u=\frac{\tan u\left(3-\tan ^{2} u\right)}{1-3 \tan ^{2} u} $$

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

Exer. 1-38: Find all solutions of the equation. $$ 4 \sin ^{2} x-3=0 $$

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

Exer. 25-36: Verify the reduction formula. $$ \sin \left(\theta-\frac{3 \pi}{2}\right)=\cos \theta $$

Use sum-to-product formulas to find the solutions of the equation. $$ \sin t+\sin 3 t=\sin 2 t $$

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

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

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

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

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

Use sum-to-product formulas to find the solutions of the equation. $$ \cos x=\cos 3 x $$

Verify the identity. $$ \tan \frac{\theta}{2}=\csc \theta-\cot \theta $$

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

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

Exer. 1-50: Verify the identity. $$ \frac{\cos ^{3} x-\sin ^{3} x}{\cos x-\sin x}=1+\sin x \cos x $$

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

Use sum-to-product formulas to find the solutions of the equation. $$ \cos 4 x-\cos 3 x=0 $$

Verify the identity. $$ \tan ^{2} \frac{\theta}{2}=1-2 \cot \theta \csc \theta+2 \cot ^{2} \theta $$

Exer. 31-32: Complete the statements. (a) As \(x \rightarrow-1^{+}, \sin ^{-1} x \rightarrow\) (b) As \(x \rightarrow 1^{-}, \cos ^{-1} x \rightarrow\) (c) As \(x \rightarrow \infty, \tan ^{-1} x \rightarrow\)

Exer. 1-38: Find all solutions of the equation. $$ (2 \sin \theta+1)(2 \cos \theta+3)=0 $$

Exer. 1-50: Verify the identity. $$ (\csc t-\cot t)^{4}(\csc t+\cot t)^{4}=1 $$

Exer. 25-36: Verify the reduction formula. $$ \cos \left(x+\frac{3 \pi}{2}\right)=\sin x $$

Express in terms of the cosine function with exponent \(1 .\) $$ \cos ^{4} \frac{\theta}{2} $$

Exer. 31-32: Complete the statements. (a) As \(x \rightarrow 1^{-}, \sin ^{-1} x \rightarrow\) (b) As \(x \rightarrow-1^{+}, \cos ^{-1} x \rightarrow\) (c) As \(x \rightarrow-\infty, \tan ^{-1} x \rightarrow\)

Exer. 1-38: Find all solutions of the equation. $$ (2 \sin u-1)(\cos u-\sqrt{2})=0 $$

Exer. 1-50: Verify the identity. $$ (a \cos t-b \sin t)^{2}+(a \sin t+b \cos t)^{2}=a^{2}+b^{2} $$

Use sum-to-product formulas to find the solutions of the equation. $$ \cos 3 x=-\cos 6 x $$

Express in terms of the cosine function with exponent \(1 .\) $$ \cos ^{4} 2 x $$

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

Exer. 1-50: Verify the identity. $$ \frac{\sin \alpha \cos \beta+\cos \alpha \sin \beta}{\cos \alpha \cos \beta-\sin \alpha \sin \beta}=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta} $$

Use sum-to-product formulas to find the solutions of the equation. $$ \sin 2 x-\sin 5 x=0 $$

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

Express in terms of the cosine function with exponent \(1 .\) $$ \sin ^{4} 2 x $$

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

Use sum-to-product formulas to find the solutions of the equation. $$ \sin 5 x-\sin x=2 \cos 3 x $$

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

Express in terms of the cosine function with exponent \(1 .\) $$ \sin ^{4} \frac{\theta}{2} $$

Exer. 1-38: Find all solutions of the equation. $$ \sin 2 x(\csc 2 x-2)=0 $$

Exer. 1-50: Verify the identity. $$ \frac{\tan \alpha}{1+\sec \alpha}+\frac{1+\sec \alpha}{\tan \alpha}=2 \csc \alpha $$

Exer. 25-36: Verify the reduction formula. $$ \tan \left(\theta+\frac{\pi}{2}\right)=-\cot \theta $$

Find the solutions of the equation that are in the interval \([0,2 \pi)\). $$ \sin 2 t+\sin t=0 $$

Exer. 33-42: Sketch the graph of the equation. $$ y=\sin ^{-1}(x-2)+\frac{\pi}{2} $$

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

Exer. 1-50: Verify the identity. $$ \frac{\csc x}{1+\csc x}-\frac{\csc x}{1-\csc x}=2 \sec ^{2} x $$

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