These identities are useful to find the exact value of the sine, cosine, or tangent at a given angle. For example, knowing that cos30∘=23, the exact value of cos15∘ can be found by using the second formula. In this case, the first step is to rewrite 15∘ as 230∘.cos15∘=cos230∘=21+cos30∘=21+23=22+3
Above, the positive sign was chosen because 15∘ lies in the first quadrant and cosine is positive there. Below, a proof of the second identity is shown. The other two identities can be proven by following a similar reasoning.
First, write the Double-Angle Identity for the cosine.
Next, solve this equation for cosx.