Express cos(θ/2) and sin(θ/2) in terms of cosθ

Question

Express $\cos(\frac{\theta}{2})$ and $\sin(\frac{\theta}{2})$ in terms of $\cos(\theta)$

My workings below ($\cos(\theta)$+ i $\sin(\theta))^\frac{1}{2}$ as $n=\frac{1}{2}$

I think the modulus equals 1 but not sure in terms of $\cos(\theta)$

Answer 1

\begin{eqnarray*} \cos \theta + i \sin \theta = (\cos (\theta /2) + i \sin (\theta /2))^2. \end{eqnarray*} Equating real & imaginary parts \begin{eqnarray*} \cos \theta = \cos^2 (\theta /2) - \sin^2 (\theta /2) \\ \sin \theta = 2\cos (\theta /2) \sin (\theta /2). \end{eqnarray*}

Answer 2

Using the identity $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ \begin{align} \cos\theta & =\cos\left(\frac\theta 2+ \frac\theta 2\right) \\ & =\cos\left(\frac\theta 2\right)\cos\left(\frac\theta 2\right)-\sin\left(\frac\theta 2\right)\sin\left(\frac\theta 2\right)\\ \cos\theta & = \cos^2\left(\frac\theta 2\right)-\sin^2\left(\frac\theta 2\right) \end{align}