The value of the determinant | array cc ^2(θ)/(2) & ^2(θ)/(2) ^2(θ)/(2) & ^2(θ)/(2) array | for all values of θ , is

Correct answer: B. θ | array cc ^2(θ)/(2) & ^2(θ)/(2) ^2(θ)/(2) & ^2(θ)/(2) array |=( ^4 (θ)/(2)- ^4 (θ)/(2))=( ^2 (θ)/(2)+ ^2 (θ)/(2))( ^2 (θ)/(2)- ^2 (θ)/(2))=(1)( 2((θ)/(2)))= θ

NDA 2017

Exam: NDA
Year: 2017
Subject: Mathematics
Chapter: Determinants

NDA 2017 · Mathematics · Determinants

The value of the determinant | array cc ^2(θ)/(2) & ^2(θ)/(2)...

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

Options

A $1$
B $\cos \theta$
C $\sin \theta$
D $\cos 2 \theta$

Correct answer

B. $\cos \theta$

Step-by-step 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

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