Q166 / 270 · NDA 2017 (Phase 2) · Mathematics

The value of the determinant $\left|\begin{array}{cc}\cos ^2 \frac{\theta}{2} & \sin ^2 \frac{\theta}{2} \\ \sin ^2 \frac{\theta}{2} & \cos ^2 \frac{\theta}{2}\end{array}\right|$ for all values of $\theta$, is

  • A
    $1$
  • Correct answer
    B
    $\cos \theta$
  • C
    $\sin \theta$
  • D
    $\cos 2 \theta$

Solution

$\left|\begin{array}{cc}\cos ^2 \frac{\theta}{2} & \sin ^2 \frac{\theta}{2} \\ \sin ^2 \frac{\theta}{2} & \cos ^2 \frac{\theta}{2}\end{array}\right|=\left(\cos ^4 \frac{\theta}{2}-\sin ^4 \frac{\theta}{2}\right)$ $=\left(\cos ^2 \frac{\theta}{2}+\sin ^2 \frac{\theta}{2}\right)\left(\cos ^2 \frac{\theta}{2}-\sin ^2 \frac{\theta}{2}\right)$ $=(1)\left(\cos 2\left(\frac{\theta}{2}\right)\right)=\cos \theta$

Hence, the correct answer is option: 'B'.

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