Question

If $\left(\begin{array}{cc}1& -\tan \theta \\ \tan \theta & 1\end{array}\right)\left(\begin{array}{cc}1& \tan \theta \\ -\tan \theta & 1\end{array}\right)=\left(\begin{array}{cc}a& -b\\ -b& a\end{array}\right)$, then the values of $a$ and $b$ are:

• A. $a=1,b=-1$
• B. $a={\sec }^2\theta ,b=0$
• C. $a=0,b={\sec }^2\theta$
• D. $a=\sin 2\theta ,b=-\cos 2\theta$

Answer 1

The correct option is B $a={\sec }^2\theta ,b=0$

$\left[\begin{array}{cc}1+{\tan }^2\theta & \tan \theta -\tan \theta \\ \tan \theta -\tan \theta & 1+{\tan }^2\theta \end{array}\right]=\left[\begin{array}{cc}a& -b\\ -b& a\end{array}\right]$

On comparing the corresponding entries, we have:
${\sec }^2\theta =a$ and $b=0$