simplify the expression by using a double - angle formula. cos²(θ/2) - sin²(θ/2)

Explanation:

Step1: Recall double - angle formula

The double - angle formula for cosine is $\cos(2\alpha)=\cos^{2}\alpha-\sin^{2}\alpha$. Let $\alpha = \frac{\theta}{2}$.

Step2: Substitute into the formula

Substituting $\alpha=\frac{\theta}{2}$ into the double - angle formula $\cos(2\alpha)=\cos^{2}\alpha-\sin^{2}\alpha$, we get $\cos^{2}\frac{\theta}{2}-\sin^{2}\frac{\theta}{2}=\cos(2\times\frac{\theta}{2})$.

Step3: Simplify the expression

$2\times\frac{\theta}{2}=\theta$, so $\cos(2\times\frac{\theta}{2})=\cos\theta$.

Answer:

$\cos\theta$