Chapter 8

Use the Addition Formula for Tangent to prove the Double-Angle Formula for Tangent.

\(25-38\) . Find all solutions of the given equation. $$ 9 \sin ^{2} \theta-1=0 $$

Verify the identity. $$ \csc x[\csc x+\sin (-x)]=\cot ^{2} x $$

\(35-38=(a)\) Graph \(f\) and \(g\) in the given viewing rectangle and find the intersection points graphically, rounded to two decimal places. (b) Find the intersection points of \(f\) and \(g\) algebraically. Give exact answers. $$ f(x)=\tan x, g(x)=\sqrt{3} ; \quad-\frac{\pi}{2}, \frac{\pi}{2} | \text { by }[-10,10] $$

Prove the identity. $$ \tan x-\tan y=\frac{\sin (x-y)}{\cos x \cos y} $$

\(37-42\) Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. $$ \sin X=\frac{3}{5}, \quad 0^{\circ} < x < 90^{\circ} $$

\(25-38\) . Find all solutions of the given equation. $$ \sec ^{2} \theta-2=0 $$

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

\(35-38=(a)\) Graph \(f\) and \(g\) in the given viewing rectangle and find the intersection points graphically, rounded to two decimal places. (b) Find the intersection points of \(f\) and \(g\) algebraically. Give exact answers. $$ f(x)=\sin x-1, g(x)=\cos x,[-2 \pi, 2 \pi] \text { by }[-2.5,1.5] $$

Prove the identity. $$ 1-\tan x \tan y=\frac{\cos (x+y)}{\cos x \cos y} $$

\(37-42\) Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. $$ \cos x=-\frac{4}{5}, \quad 180^{\circ} < x < 270^{\circ} $$

\(25-38\) . Find all solutions of the given equation. $$ \csc ^{2} \theta-4=0 $$

Verify the identity. $$ (\sin x+\cos x)^{2}=1+2 \sin x \cos x $$

\(39-42\) . Use an Addition or Subtraction Formula to simplify the equation. Then find all solutions in the interval \([0,2 \pi) .\) \(\cos \theta \cos 3 \theta-\sin \theta \sin 3 \theta=0\)

Prove the identity. $$ \frac{\sin (x+y)-\sin (x-y)}{\cos (x+y)+\cos (x-y)}=\tan y $$

\(37-42\) Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. $$ \csc x=3, \quad 90^{\circ} < x < 180^{\circ} $$

\(39-56 \approx\) Solve the given equation. $$ \left(\tan ^{2} \theta-4\right)(2 \cos \theta+1)=0 $$

Verify the identity. $$ (1-\cos \beta)(1+\cos \beta)=\frac{1}{\csc ^{2} \beta} $$

\(39-42\) = Use an Addition or Subtraction Formula to simplify the equation. Then find all solutions in the interval \([0,2 \pi)\) \(\cos \theta \cos 2 \theta+\sin \theta \sin 2 \theta=\frac{1}{2}\)

Prove the identity. $$ \cos (x+y) \cos (x-y)=\cos ^{2} x-\sin ^{2} y $$

\(37-42\) Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. $$ \tan x=1, \quad 0^{\circ} < x < 90^{\circ} $$

\(39-56 \approx\) Solve the given equation. $$ (\tan \theta-2)\left(16 \sin ^{2} \theta-1\right)=0 $$

Verify the identity. $$ \frac{\cos x}{\sec x}+\frac{\sin x}{\csc x}=1 $$

Prove the identity. $$ \begin{aligned} \sin (x+y+z)=& \sin x \cos y \cos z+\cos x \sin y \cos z \\\ &+\cos x \cos y \sin z-\sin x \sin y \sin z \end{aligned} $$

\(37-42\) Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. $$ \sec x=\frac{3}{2}, \quad 270^{\circ} < x < 360^{\circ} $$

\(39-56 \approx\) Solve the given equation. $$ 4 \cos ^{2} \theta-4 \cos \theta+1=0 $$

Verify the identity. $$ \frac{(\sin x+\cos x)^{2}}{\sin ^{2} x-\cos ^{2} x}=\frac{\sin ^{2} x-\cos ^{2} x}{(\sin x-\cos x)^{2}} $$

\(39-42\) . Use an Addition or Subtraction Formula to simplify the equation. Then find all solutions in the interval \([0,2 \pi) .\) \(\sin 3 \theta \cos \theta-\cos 3 \theta \sin \theta=0\)

Prove the identity. $$ \begin{aligned} \tan (x-y)+\tan (y-z) &+\tan (z-x) \\ &=\tan (x-y) \tan (y-z) \tan (z-x) \end{aligned} $$

\(37-42\) Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. $$ \cot x=5, \quad 180^{\circ} < x < 270^{\circ} $$

\(39-56 \approx\) Solve the given equation. $$ 2 \sin ^{2} \theta-\sin \theta-1=0 $$

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

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

\(43-46\). Write the given expression as an algebraic expression in \(x\). $$ \sin \left(2 \tan ^{-1} x\right) $$

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

\(39-56 \approx\) Solve the given equation. $$ 3 \sin ^{2} \theta-7 \sin \theta+2=0 $$

Verify the identity. $$ \frac{\sec t-\cos t}{\sec t}=\sin ^{2} t $$

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

\(43-46\). Write the given expression as an algebraic expression in \(x\). $$ \tan \left(2 \cos ^{-1} x\right) $$

\(39-56 \approx\) Solve the given equation. $$ \tan ^{4} \theta-13 \tan ^{2} \theta+36=0 $$

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

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

\(43-46\). Write the given expression as an algebraic expression in \(x\). $$ \sin \left(\frac{1}{2} \cos ^{-1} x\right) $$

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

\(39-56 \approx\) Solve the given equation. $$ 2 \cos ^{2} \theta-7 \cos \theta+3=0 $$

Verify the identity. $$ \frac{1}{1-\sin ^{2} y}=1+\tan ^{2} y $$

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

\(43-46\). Write the given expression as an algebraic expression in \(x\). $$ \cos \left(2 \sin ^{-1} x\right) $$

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

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