Sum & Difference Identities

For any angles or values \(A\) and \(B\), the following relationships are always true.

\[\begin{align*} \sin(A + B) &= \sin(A)\cos(B) + \cos(A)\sin(B) \\[4pt] \sin(A - B) &= \sin(A)\cos(B) - \cos(A)\sin(B) \\[16pt] \cos(A + B) &= \cos(A)\cos(B) - \sin(A)\sin(B) \\[4pt] \cos(A - B) &= \cos(A)\cos(B) + \sin(A)\sin(B) \\[16pt] \tan(A + B) &= \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)} \\[4pt] \tan(A - B) &= \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)} \end{align*}\]

Simplifying Expressions

As we add in a new set of identities into the mix, remember that the overall process is still the same. Ask yourself, is there an identity I can use or algebra I can do? These new identities allow us to simplify more types of expressions, but they also help us compute the trig function values of angles other than \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), or \(\frac{\pi}{2}\). Let's look at some examples that use these new identities.

Verify Identities

Just like simplifying, the process to verify an identity hasn't changed even though we have more identities. Let's look at a few examples.

Trigonometric Identities