Chapter 7

State whether or not the equation is an identity. If it is an identity, prove it. $$\cot ^{2} x-1=\csc ^{2} x$$

Find the exact functional value without using a calculator. $$\cos \left[\tan ^{-1}(-3 / 4)\right]$$

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

State whether or not the equation is an identity. If it is an identity, prove it. $$\left(\cos ^{2} x-1\right)\left(\tan ^{2} x+1\right)=-\tan ^{2} x$$

Write each expression as a product. $$\cos 2 x+\cos 6 x$$

Find the exact functional value without using a calculator. $$\cos \left[\sin ^{-1}(12 / 13)\right]$$

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

State whether or not the equation is an identity. If it is an identity, prove it. $$\left(1-\cos ^{2} x\right) \csc x=\sin x$$

Find the exact functional value without using a calculator. $$\tan \left[\cos ^{-1}(5 / 13)\right]$$

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

State whether or not the equation is an identity. If it is an identity, prove it. $$\tan x=\frac{\sec x}{\csc x}$$

Write each expression as a product. $$\cos 5 x-\cos 7 x$$

Find the exact functional value without using a calculator. $$\sin \left[\tan ^{-1}(12 / 5)\right]$$

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

State whether or not the equation is an identity. If it is an identity, prove it. $$\frac{\cos (-x)}{\sin (-x)}=-\cot x$$

Find the exact functional value without using a calculator. $$\cos \left[\sin ^{-1}(\sqrt{3} / 5)\right]$$

Use an appropriate substitution (as in Example 7 ) to find all solutions of the equation. $$\sin 2 x=-\sqrt{3} / 2$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\cos ^{4} x-\sin ^{4} x=\cos ^{2} x-\sin ^{2} x$$

Find the exact functional value without using a calculator. $$\tan \left[\sin ^{-1}(\sqrt{7} / 12)\right]$$

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

Use an appropriate substitution (as in Example 7 ) to find all solutions of the equation. $$\cos 2 x=\sqrt{2} / 2$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\cot ^{2} x-\cos ^{2} x=\cos ^{2} x \cot ^{2} x$$

Assume sin \(x=.6\) and \(0

Find the exact functional value without using a calculator. $$\sin \left[\cos ^{-1}(3 / \sqrt{13})\right]$$

If \(x\) is in the first and \(y\) is in the second quadrant, \(\sin x=24 / 25,\) and \(\sin y=4 / 5,\) find the exact value of \(\sin (x+y)\) and \(\tan (x+y)\) and the quadrant in which \(x+y\) lies.

Use an appropriate substitution (as in Example 7 ) to find all solutions of the equation. $$2 \cos \frac{x}{2}=\sqrt{2}$$

Assume sin \(x=.6\) and \(0

Find the exact functional value without using a calculator. $$\tan \left[\cos ^{-1}(8 / 9)\right]$$

If \(x\) and \(y\) are in the second quadrant, \(\sin x=1 / 3,\) and \(\cos y=-3 / 4,\) find the exact value of \(\sin (x+y)\) \(\cos (x+y), \tan (x+y),\) and find the quadrant in which \(x+y\) lies.

Use an appropriate substitution (as in Example 7 ) to find all solutions of the equation. $$2 \sin \frac{x}{3}=1$$

State whether or not the equation is an identity. If it is an identity, prove it. $$(1+\tan x)^{2}=\sec ^{2} x$$

Find the exact functional value without using a calculator. $$\sin \left[\tan ^{-1}(\sqrt{5} / 10)\right]$$

If \(x\) is in the first and \(y\) is in the second quadrant, \(\sin x=4 / 5,\) and \(\cos y=-12 / 13,\) find the exact value of \(\cos (x+y)\) and \(\tan (x+y)\) and the quadrant in which \(x+y\) lies.

Use an appropriate substitution (as in Example 7 ) to find all solutions of the equation. $$\tan 3 x=-\sqrt{3}$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\frac{1+\sin x}{\sin x}=\frac{\cot ^{2} x}{\csc x-1}$$

Assume sin \(x=.6\) and \(0

Find the exact functional value without using a calculator. $$\cos \left[\tan ^{-1}(3 / 7)\right]$$

If \(x\) is in the fourth and \(y\) is in the first quadrant, \(\cos x=1 / 3,\) and \(\cos y=2 / 3,\) find the exact value of \(\sin (x-y)\) and \(\tan (x-y)\) and the quadrant in which \(x-y\) lies.

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

Express \(\sin (u+v+w)\) in terms of sines and cosines of \(u, v,\) and \(w .\) IHint: First apply the addition identity with \(x=u+v \text { and } y=w .]\)

Use an appropriate substitution (as in Example 7 ) to find all solutions of the equation. $$5 \cos 3 x=-3$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\frac{1+\sin x}{1-\sin x}=\frac{\sec x+\tan x}{\sec x-\tan x}$$

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

Express \(\cos (x+y+z)\) in terms of sines and cosines of \(x, y,\) and \(z\)

Use an appropriate substitution (as in Example 7 ) to find all solutions of the equation. $$2 \tan 4 x=16$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\frac{\sin x}{\cos x}+\frac{\cos x}{1+\sin x}=\sec x$$

Express \(\cos 3 x\) in terms of \(\cos x\)

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

$$\text { If } x+y=\pi / 2, \text { show that } \sin ^{2} x+\sin ^{2} y=1$$

Use an appropriate substitution (as in Example 7 ) to find all solutions of the equation. $$4 \tan \frac{x}{2}=8$$