How do you prove a trig identity?

 
Relevant MM Lessons:This question involves a few major concepts across several lessons - for a deeper dive and for a full gambit of practice and common tricks, make sure to check out each Relevant Mister Math Lesson in the list above.

Identity Proofs

The process of proving trig identities doesn't have a lot of fans out there in the world. It's more like puzzle-solving than math, and doesn't accomplish anything real. After all, identity means "itself", so when you are asked to prove a trig identity, you are being asked to show something is itself, which inherently sounds pointless.Regardless, we've gotta get it done to get the grade! Here are some quick rules and tips to follow.Rules
  • On each side of the equals sign, use basic facts, trig relationships, and algebra rules to change the initial expressions and show that they are indeed equal.
  • Never perform an operation to both sides of the equation (such as multiplying both sides by $\sin(x)$). Only perform manipulations and changes.
Keep in mind also that some teachers, for some reason, demand that you only pick one side of the identity to work on and that you absolutely do not touch or change the other side. I think this is ludicrous but I've seen it a few times.Tips
  • Turn functions into sine and cosine functions
  • Obtain common denominators and combine terms if you have fractions
  • Look for $\sin^2$ and $\cos^2$ terms
  • Keep your goal in mind
I'll show a quick example of what I mean by each of these. Remember to check out the two lessons on trig identity proofs (basic identity proofs » and advanced identity proofs ») for actual practice - which is unfortunately the only way to master these things.

Turn Stuff Into sin and cos

Consider the following identity to prove:While it is not the only step in the proof, it is a helpful first one.

Obtain Common Denominators

If you know you need to make one side into a single fraction (based on what you see on the other side), you'll need to get common denominators. It's usually best to do this earlier rather than later.Consider the following identity to prove:Multiply each fraction on the left to get a common denominator.

Look for Squared sin and cos Terms

Very frequently, identities will require the use of the common pythagorean relationship$$\sin^2(x) + \cos^2(x) = 1$$which can be arranged as$$\sin^2(x) = 1 - \cos^2(x)$$or$$\cos^2(x) = 1 - \sin^2(x)$$Whenever you encounter squared sine or cosine terms, applying this identity is often helpful. Consider the following identity to prove:$$\frac{\sin^2 (x)}{\sin(x)(\cos(x)} + \frac{\cos^2(x)}{\sin(x)\cos(x)} = \frac{2}{\csc(2x)}$$Since we already have common denominators, we can combine the pieces, which will leave a Pythagorean identity in the numerator.$$\frac{\sin^2(x) + \cos^2(x)}{\sin(x)(\cos(x)} = \frac{2}{\csc(2x)}$$$$\frac{1}{\sin(x)(\cos(x)} = \frac{2}{\csc(2x)}$$

Keep Your Goal In Mind

Consider the following identity to prove:$$\frac{2\tan(\theta)}{1+\tan^2(\theta)} = \sin(2\theta)$$Practice and intuition will tell you to focus on working on the left side. However, do not lose sight of the right side that you seek to obtain! It's a single object, so your actions on the left should work toward the goal of converting the fraction you are starting with into an expression that is not a fraction.