The half-angle identities are special cases of angle difference identities. To evaluate trigonometric functions of half an angle, the following identities can be applied. The sign of each formula is determined by the quadrant in which $\frac{\theta }{2}$ lies.

These identities are useful to find the exact value of the sine, cosine, or tangent at a given angle. For example, knowing that $\cos 30^{\circ }=\frac{\sqrt{3}}{2},$ the exact value of $\cos 15^{\circ }$ can be found by using the second formula. In this case, the first step is to rewrite $15^{\circ }$ as $\frac{30^{\circ }}{2}.$ $$\begin{array}{rl}\cos 15^{\circ }& =\cos \frac{30^{\circ }}{2}\\ & =\sqrt{\frac{1+\cos 30^{\circ }}{2}}\\ & =\sqrt{\frac{1+\frac{\sqrt{3}}{2}}{2}}=\frac{\sqrt{2+\sqrt{3}}}{2}\end{array}$$ Above, the positive sign was chosen because $15^{\circ }$ 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.