Question

Express $\sin \frac{x}{2}$ in terms of $\cos x$ using the double angle identity.

Answer 1

Using properties of trigonometric functions

$$\begin{gathered} \Rightarrow \cos 2x=1-2{\sin }^{2}x \\ \Rightarrow \cos x=1-2{\sin }^2\frac{x}{2} \\ \Rightarrow 2{\sin }^2\frac{x}{2}=1-\cos x \\ \Rightarrow {\sin }^2\frac{x}{2}=\frac{1-\cos x}{2} \\ \Rightarrow \sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}} \end{gathered}$$

Hence, $\sin \frac{x}{2}$ in terms of $\cos x$ can be expressed as $\pm \sqrt{\frac{1-\cos x}{2}}$.