Chapter 7

(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. $$\begin{array}{l}f(x)=3 \cos x+1, g(x)=\cos x-1 \\ [-2 \pi, 2 \pi] \text { by }[-2.5,4.5] \end{array}$$

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

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

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

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

(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. $$\begin{aligned} &f(x)=\sin 2 x+1, g(x)=2 \sin 2 x+1\\\ &[-2 \pi, 2 \pi] \text { by }[-1.5,3.5] \end{aligned}$$

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

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

Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. $$\sin x=\frac{3}{3}, \quad 0^{\circ}

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

(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. $$f(x)=\tan x, g(x)=\sqrt{3} ;\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \text { by }[-10,10]$$

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

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

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}

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

(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. $$f(x)=\sin x-1, g(x)=\cos x ;[-2 \pi, 2 \pi] \text { by }[-2.5,1.5]$$

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

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

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

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

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

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

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

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

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

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

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

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

Solve the given equation. $$4 \cos ^{2} \theta-4 \cos \theta+1=0$$

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

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

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

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

Solve the given equation. $$2 \sin ^{2} \theta-\sin \theta-1=0$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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