Section 1
Verifying Identities
In Exercise $1-4,$ completely factor each expression.$$4 x^{2}-9$$
In Exercise $1-4,$ completely factor each expression.$$x^{2}+4 x+4$$
In Exercise $1-4,$ completely factor each expression.$$9 x^{2}-81$$
In Exercise $1-4,$ completely factor each expression.$$2 x^{2}-16 x+32$$
In Exercises $5-8,$ completely simplify each expression.$$\frac{2}{1+x}-\frac{1}{1-x}$$
In Exercises $5-8,$ completely simplify each expression.$$\frac{3}{x-2}+\frac{4}{x+2}$$
Completely simplify each expression.$$\frac{5}{x-1}+\frac{3}{1-x}$$
In Exercises $5-8,$ completely simplify each expression.$$\frac{7}{x-2}+\frac{4}{2-x}$$
In Exercises $9-14$, show that the given equations are not identities.$$\cos x=0.5$$
In Exercises $9-14$, show that the given equations are not identities.$$\tan x=1$$
In Exercises $9-14$, show that the given equations are not identities.$$\sin (x+\pi)=\sin x+\sin \pi$$
In Exercises $9-14$, show that the given equations are not identities.$$\cos \left(x-\frac{\pi}{2}\right)=\cos x-\cos \frac{\pi}{2}$$
In Exercises $9-14$, show that the given equations are not identities.$$\sin 2 x=2 \sin x$$
In Exercises $9-14$, show that the given equations are not identities.$$\cos 2 x=2 \cos x$$
In Exercises $15-20,$ write each expression in terms of $\sin x$ and/or $\cos x$ only.$$\cot x \csc x$$
In Exercises $15-20,$ write each expression in terms of $\sin x$ and/or $\cos x$ only.$$\csc ^{2} x$$
In Exercises $15-20,$ write each expression in terms of $\sin x$ and/or $\cos x$ only.$$\frac{\tan ^{2} x}{\sin x}$$
In Exercises $15-20,$ write each expression in terms of $\sin x$ and/or $\cos x$ only.$$\sec x \cot x$$
In Exercises $15-20,$ write each expression in terms of $\sin x$ and/or $\cos x$ only.$$\sec ^{2} x-1$$
In Exercises $15-20,$ write each expression in terms of $\sin x$ and/or $\cos x$ only.$$1-\tan x$$
In Exercises $21-26,$ factor the given trigonometric expressions completely.$$\sin x+\sin x \cos x$$
In Exercises $21-26,$ factor the given trigonometric expressions completely.$$\tan ^{2} x-\sec x \tan x$$
In Exercises $21-26,$ factor the given trigonometric expressions completely.$$1-\sin ^{2} x$$
In Exercises $21-26,$ factor the given trigonometric expressions completely.$$\sec ^{2} x-\tan ^{2} x$$
In Exercises $21-26,$ factor the given trigonometric expressions completely.$$\sin ^{2} x-\cos ^{2} x$$
In Exercises $21-26,$ factor the given trigonometric expressions completely.$$\sin ^{4} x-1$$
In Exercises $27-80,$ verify the given identities.$$\tan x \csc x=\sec x$$
In Exercises $27-80,$ verify the given identities.$$\cot x \sec x=\csc x$$
In Exercises $27-80,$ verify the given identities.$$\sin x+\cos x \cot x=\csc x$$
In Exercises $27-80,$ verify the given identities.$$\cos x+\tan x \sin x=\sec x$$
In Exercises $27-80,$ verify the given identities.$$\sec ^{2} x\left(1-\sin ^{2} x\right)=1$$
In Exercises $27-80,$ verify the given identities.$$\csc ^{2} x\left(1-\cos ^{2} x\right)=1$$
In Exercises $27-80,$ verify the given identities.$$\sin ^{2}(-x)+\cos ^{2}(-x)=1$$
In Exercises $27-80,$ verify the given identities.$$\sin ^{2}(-x)+\cos ^{2} x=1$$
In Exercises $27-80,$ verify the given identities.$$\left(\sec ^{2} x-1\right) \cot ^{2} x=1$$
In Exercises $27-80,$ verify the given identities.$$\left(\csc ^{2} x-1\right) \tan ^{2} x=1$$
In Exercises $27-80,$ verify the given identities.$$\frac{\cot x}{\csc x}=\cos x$$
In Exercises $27-80,$ verify the given identities.$$\frac{\tan x}{\sec x}=\sin x$$
In Exercises $27-80,$ verify the given identities.$$\sin ^{2} x \sec x=\sec x-\cos x$$
In Exercises $27-80,$ verify the given identities.$$\cos ^{2} x \csc x=\csc x-\sin x$$
In Exercises $27-80,$ verify the given identities.$$(\cos x+\sin x)^{2}-2 \sin x \cos x=1$$
In Exercises $27-80,$ verify the given identities.$$(\cos x-\sin x)^{2}+2 \sin x \cos x=1$$
In Exercises $27-80,$ verify the given identities.$$\sec x+\tan x=\frac{1+\sin x}{\cos x}$$
In Exercises $27-80,$ verify the given identities.$$\csc x+\cot x=\frac{\sin x}{1-\cos x}$$
In Exercises $27-80,$ verify the given identities.$$\cos ^{3} x=\cos x-\cos x \sin ^{2} x$$
In Exercises $27-80,$ verify the given identities.$$\sin ^{3} x=\sin x-\sin x \cos ^{2} x$$
In Exercises $27-80,$ verify the given identities.$$\sec x \cos ^{3} x=1-\sin ^{2} x$$
In Exercises $27-80,$ verify the given identities.$$\sec ^{3} x=\frac{1+\tan ^{2} x}{\cos x}$$
In Exercises $27-80,$ verify the given identities.$$\frac{1}{1+\cos x}+\frac{1}{1-\cos x}=2 \csc ^{2} x$$
In Exercises $27-80,$ verify the given identities.$$\frac{1}{1+\cos x}+\frac{1}{1-\cos x}=-2 \csc ^{2} x \cos x$$
In Exercises $27-80,$ verify the given identities.$$\frac{\sin x-\sin (-x)}{1-\cos ^{2} x}=2 \csc x$$
In Exercises $27-80,$ verify the given identities.$$\frac{\cos x+\cos (-x)}{1-\sin ^{2} x}=2 \sec x$$
In Exercises $27-80,$ verify the given identities.$$\frac{\sec ^{2} x}{1+\sin x}=\frac{\sec ^{2} x-\sec x \tan x}{\cos ^{2} x}$$
In Exercises $27-80,$ verify the given identities.$$\frac{\csc ^{2} x}{1+\cos x}=\frac{\csc ^{2} x-\csc x \cot x}{\sin ^{2} x}$$
In Exercises $27-80,$ verify the given identities.$$\cos ^{2} x-\sin ^{2} x=1-2 \sin ^{2} x$$
In Exercises $27-80,$ verify the given identities.$$\sec ^{2} x+\tan ^{2} x=1+2 \tan ^{2} x$$
In Exercises $27-80,$ verify the given identities.$$\tan x+\cot x=\sec x \csc x$$
In Exercises $27-80,$ verify the given identities.$$\tan x-\cot x=\sec x \csc x-2 \cos x \csc x$$
In Exercises $27-80,$ verify the given identities.$$\cos ^{4} x-\sin ^{4} x=\cos ^{2} x-\sin ^{2} x$$
In Exercises $27-80,$ verify the given identities.$$\cot ^{2} x-\tan ^{2} x=\csc ^{2} x-\sec ^{2} x$$
In Exercises $27-80,$ verify the given identities.$$\frac{\tan ^{2} x-1}{1+\tan ^{2} x}=2 \sin ^{2} x-1$$
In Exercises $27-80,$ verify the given identities.$$\frac{\cot ^{2} x-1}{1+\cot ^{2} x}=2 \cos ^{2} x-1$$
In Exercises $27-80,$ verify the given identities.$$\csc ^{2} x+\sec ^{2} x=\csc ^{2} x \sec ^{2} x$$
In Exercises $27-80,$ verify the given identities.$$1-\tan ^{4} x=\sec ^{2} x\left(1-\tan ^{2} x\right)$$
In Exercises $27-80,$ verify the given identities.$$\sec ^{4} x-\tan ^{4} x=\sec ^{2} x+\tan ^{2} x$$
In Exercises $27-80,$ verify the given identities.$$1-\sin ^{4} x=\cos ^{2} x+\cos ^{2} x \sin ^{2} x$$
In Exercises $27-80,$ verify the given identities.$$\frac{1}{1+\cos x}=\csc ^{2} x-\cot x \csc x$$
In Exercises $27-80,$ verify the given identities.$$\frac{1}{1-\sin x}=\sec ^{2} x+\tan x \sec x$$
In Exercises $27-80,$ verify the given identities.$$(\csc x-\cot x)^{2}=\frac{1-\cos x}{1+\cos x}$$
In Exercises $27-80,$ verify the given identities.$$(\sec x-\tan x)^{2}=\frac{1-\sin x}{1+\sin x}$$
In Exercises $27-80,$ verify the given identities.$$\cot x+\tan x=\sec ^{2} x \cot x$$
In Exercises $27-80,$ verify the given identities.$$\sin x+\cos x+\frac{\sin x}{\cot x}=\sec x+\csc x-\frac{\cos x}{\tan x}$$
In Exercises $27-80,$ verify the given identities.$$\frac{\sec ^{2} x-1}{\tan x}=\frac{\sec x}{\csc x}$$
In Exercises $27-80,$ verify the given identities.$$\frac{\csc ^{2} x-1}{\cot x}=\frac{\csc x}{\sec x}$$
In Exercises $27-80,$ verify the given identities.$$\frac{\sin x}{\csc x-\cot x}=1+\cos x$$
In Exercises $27-80,$ verify the given identities.$$\frac{\cos x-\sin x}{\cos x+\sin x}=\frac{\cot x-1}{\cot x+1}$$
In Exercises $27-80,$ verify the given identities.$$a \csc ^{2} x(1+\cos x)(1-\cos x)=a$$
In Exercises $27-80,$ verify the given identities.$$b(\sec x-\tan x)(\sec x+\tan x)=b$$
In Exercises $27-80,$ verify the given identities.$$\ln |\tan x|=-\ln |\cot x|$$
In Exercises $27-80,$ verify the given identities.$$\ln |\sec x|=-\ln |\cos x|$$
In Exercises $81-86,$ use a graphing utility to graph each side of the equation and decide whether the equation is an identity. You need not verify the ones that are identities.$$\sin (x+\pi)=\sin x+\pi$$
In Exercises $81-86,$ use a graphing utility to graph each side of the equation and decide whether the equation is an identity. You need not verify the ones that are identities.$$\cos 2 x=2 \cos x$$
In Exercises $81-86,$ use a graphing utility to graph each side of the equation and decide whether the equation is an identity. You need not verify the ones that are identities.$$\cos 2 x=1-2 \sin ^{2} x$$
In Exercises $81-86,$ use a graphing utility to graph each side of the equation and decide whether the equation is an identity. You need not verify the ones that are identities.$$\sin 2 x=2 \sin x \cos x$$
In Exercises $81-86,$ use a graphing utility to graph each side of the equation and decide whether the equation is an identity. You need not verify the ones that are identities.$$\sin (x-\pi)=\sin x$$
In Exercises $81-86,$ use a graphing utility to graph each side of the equation and decide whether the equation is an identity. You need not verify the ones that are identities.$$\cos (x+\pi)=-\cos x$$
Explain why $\sin x=\sqrt{1-\cos ^{2} x}$ is not an identity.
What values of $x$ satisfy the equation $\sin x=\cos x ?$
Does the identity $\tan x \cos x=\sin x$ hold for $x=\frac{\pi}{2} ?$ Why or why not?
Does the identity csc $x \sin x=1$ hold for all real values of $x ?$ Why or why not?
Use a graphing utility to find an equivalent expression for $\cos \left(x-\frac{\pi}{2}\right)$.