Verify that
Guidance
This concept continues where the previous one left off. Now that you are comfortable simplifying expressions, we will extend the idea to verifying entire identities. Here are a few helpful hints to verify an identity:
 Change everything into terms of sine and cosine.
 Use the identities when you can.
 Start with simplifying the lefthand side of the equation, then, once you get stuck, simplify the righthand side. As long as the two sides end up with the same final expression, the identity is true.
Example A
Verify the identity
Solution: Rather than have an equal sign between the two sides of the equation, we will draw a vertical line so that it is easier to see what we do to each side of the equation. Start with changing everything into sine and cosine.
Now, it looks like we are at an impasse with the lefthand side. Let’s combine the righthand side by giving them same denominator.
The two sides reduce to the same expression, so we can conclude this is a valid identity. In the last step, we used the Pythagorean Identity,
There are usually more than one way to verify a trig identity. When proving this identity in the first step, rather than changing the cotangent to
Example B
Verify the identity
Solution:
Multiply the lefthand side of the equation by
The two sides are the same, so we are done.
Example C
Verify the identity
Solution: Change secant to cosine.
From the Negative Angle Identities, we know that
Concept Problem Revisit Start by simplifying the lefthand side of the equation.
Now simplify the righthand side of the equation. By manipulating the Trigonometric Identity,
Guided Practice
Verify the following identities.
1.
2.
3.
Answers
1. Change secant to cosine.
2. Use the identity
3. Here, start with the Negative Angle Identities and multiply the top and bottom by
Explore More
Verify the following identities.

cot(−x)=−cotx 
csc(−x)=−cscx 
tanxcscxcosx=1 
sinx+cosxcotx=cscx 
csc(π2−x)=secx 
tan(π2−x)=tanx 
cscxsinx−cotxtanx=1 
tan2xtan2x+1=sin2x 
(sinx−cosx)2+(sinx+cosx)2=2 
sinx−sinxcos2x=sin3x 
tan2x+1+tanxsecx=1+sinxcos2x 
cos2x=cscxcosxtanx+cotx 
11−sinx−11+sinx=2tanxsecx 
csc4x−cot4x=csc2x+cot2x 
(sinx−tanx)(cosx−cotx)=(sinx−1)(cosx−1)