The sum and difference formulas for tangent are very useful if you want to prove some basic trig identities. For example, you can prove the co-function identities by using the difference formula and the periodicity identities by using the sum formula. If you see a sum or a difference inside a tangent function, you can try the appropriate formula to simplify things.
For instance, you can prove this identity,
with the following steps:
Look for identities for which you can substitute.
On the left side of the equal sign, you may use the sum identity for tangent:
Working on the left side only gives you the following equation:
Use any applicable unit circle values to simplify the proof.The whole unit circle
From the unit circle, you see that
so you can plug in that value to get this:
From there, simple multiplication gives you the result that proves the equality: