The value of the determinant for all values of is ........

Mathematics (NDA) Matrices and Determinants

The value of the determinant

$\begin{bmatrix} cos^{2}\frac{\theta}{2} &sin^{2}\frac{\theta}{2} \\ sin^{2}\frac{\theta}{2} & cos^{2}\frac{\theta}{2} \end{bmatrix}$

for all values of $\theta$ ,is

cos $\theta$

sin $\theta$

cos $2\theta$

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Explanation:

$\begin{bmatrix} cos^{2}\frac{\theta}{2} &sin^{2}\frac{\theta}{2} \\ sin^{2}\frac{\theta}{2} & cos^{2}\frac{\theta}{2} \end{bmatrix}=(cos^{4}\frac{\theta}{2}-sin^{4}\frac{\theta}{2})$
$=(cos^{2}\frac{\theta}{2}+sin^{2}\frac{\theta}{2})(cos^{2}\frac{\theta}{2}-sin^{2}\frac{\theta}{2})=cos\theta$