💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!
Section 1
Solving Trigonometric Equations with Identities
We know $g(x)=\cos x$ is an even function, and $f(x)=\sin x$ and $h(x)=\tan x$ are odd functions. What about $G(x)=\cos ^{2} x, F(x)=\sin ^{2} x,$ and $H(x)=\tan ^{2} x ?$ Are they even, odd, or neither? Why?
Examine the graph of $f(x)=\sec x$ on the interval $[-\pi, \pi] .$ How can we tell whether the function is even or odd by only observing the graph of $f(x)=\sec x ?$
After examining the reciprocal identity for $\sec t$ explain why the function is undefined at certain points.
All of the Pythagorean identities are related. Describe how to manipulate the equations to get from $\sin ^{2} t+\cos ^{2} t=1$ to the other forms.
For the following exercises, use the fundamental identities to fully simplify the expression.$$\sin x \cos x \sec x$$
For the following exercises, use the fundamental identities to fully simplify the expression.$$\sin (-x) \cos (-x) \csc (-x)$$
For the following exercises, use the fundamental identities to fully simplify the expression.$$\tan x \sin x+\sec x \cos ^{2} x$$
For the following exercises, use the fundamental identities to fully simplify the expression.$$\csc x+\cos x \cot (-x)$$
For the following exercises, use the fundamental identities to fully simplify the expression.$$\frac{\cot t+\tan t}{\sec (-t)}$$
For the following exercises, use the fundamental identities to fully simplify the expression.$$3 \sin ^{3} t \csc t+\cos ^{2} t+2 \cos (-t) \cos t$$
For the following exercises, use the fundamental identities to fully simplify the expression.$$-\tan (-x) \cot (-x)$$
For the following exercises, use the fundamental identities to fully simplify the expression.$$\frac{-\sin (-x) \cos x \sec x \csc x \tan x}{\cot x}$$
For the following exercises, use the fundamental identities to fully simplify the expression.$$\frac{1+\tan ^{2} \theta}{\csc ^{2} \theta}+\sin ^{2} \theta+\frac{1}{\sec ^{2} \theta}$$
For the following exercises, use the fundamental identities to fully simplify the expression.$$\left(\frac{\tan x}{\csc ^{2} x}+\frac{\tan x}{\sec ^{2} x}\right)\left(\frac{1+\tan x}{1+\cot x}\right)-\frac{1}{\cos ^{2} x}$$
For the following exercises, use the fundamental identities to fully simplify the expression.$$\frac{1-\cos ^{2} x}{\tan ^{2} x}+2 \sin ^{2} x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$\frac{\tan x+\cot x}{\csc x} ; \cos x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$\frac{\sec x+\csc x}{1+\tan x} ; \sin x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$\frac{\cos x}{1+\sin x}+\tan x ; \cos x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$\frac{1}{\sin x \cos x}-\cot x ; \cot x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$\frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x} ; \csc x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$(\sec x+\csc x)(\sin x+\cos x)-2-\cot x ; \tan x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$\frac{1}{\csc x-\sin x} ; \sec x \text { and } \tan x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$\frac{1-\sin x}{1+\sin x}-\frac{1+\sin x}{1-\sin x} ; \sec x \text { and } \tan x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$\tan x ; \sec x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$\sec x ; \cot x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$\sec x ; \sin x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$\cot x ; \sin x$$
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.$$\cot x ; \csc x$$
For the following exercises, verify the identity.$$\cos x-\cos ^{3} x=\cos x \sin ^{2} x$$
For the following exercises, verify the identity.$$\cos x(\tan x-\sec (-x))=\sin x-1$$
For the following exercises, verify the identity.$$\frac{1+\sin ^{2} x}{\cos ^{2} x}=\frac{1}{\cos ^{2} x}+\frac{\sin ^{2} x}{\cos ^{2} x}=1+2 \tan ^{2} x$$
For the following exercises, verify the identity.$$(\sin x+\cos x)^{2}=1+2 \sin x \cos x$$
For the following exercises, verify the identity.$$\cos ^{2} x-\tan ^{2} x=2-\sin ^{2} x-\sec ^{2} x$$
For the following exercises, prove or disprove the identity.$$\frac{1}{1+\cos x}-\frac{1}{1-\cos (-x)}=-2 \cot x \csc x$$
For the following exercises, prove or disprove the identity.$$\csc ^{2} x\left(1+\sin ^{2} x\right)=\cot ^{2} x$$
For the following exercises, prove or disprove the identity.$$\left(\frac{\sec ^{2}(-x)-\tan ^{2} x}{\tan x}\right)\left(\frac{2+2 \tan x}{2+2 \cot x}\right)-2 \sin ^{2} x=\cos 2 x$$
For the following exercises, prove or disprove the identity.$$\frac{\tan x}{\sec x} \sin (-x)=\cos ^{2} x$$
For the following exercises, prove or disprove the identity.$$\frac{\sec (-x)}{\tan x+\cot x}=-\sin (-x)$$
For the following exercises, prove or disprove the identity.$$\frac{1+\sin x}{\cos x}=\frac{\cos x}{1+\sin (-x)}$$
For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.$$\frac{\cos ^{2} \theta-\sin ^{2} \theta}{1-\tan ^{2} \theta}=\sin ^{2} \theta$$
For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.$$3 \sin ^{2} \theta+4 \cos ^{2} \theta=3+\cos ^{2} \theta$$
For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.$$\frac{\sec \theta+\tan \theta}{\cot \theta+\cos \theta}=\sec ^{2} \theta$$