First slide

Applications of determinant

Question

$[\begin{matrix} 1 & - \tan \frac{θ}{2} \\ \tan \frac{θ}{2} & 1 \end{matrix}] {[\begin{matrix} 1 & \tan \frac{θ}{2} \\ - \tan \frac{θ}{2} & 1 \end{matrix}]}^{- 1} =$

Moderate

A

$[\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}]$
B

$[\begin{matrix} - \cos θ & \sin θ \\ \sin θ & \cos θ \end{matrix}]$
C

$[\begin{matrix} \sin θ & - \cos θ \\ \cos θ & \sin θ \end{matrix}]$
D

$[\begin{matrix} - \sin θ & \cos θ \\ \cos θ & - \sin θ \end{matrix}]$

Solution

$[\begin{matrix} 1 & - \tan θ / 2 \\ \tan θ / 2 & 1 \end{matrix}] \frac{1}{\sec^{2} θ / 2} [\begin{matrix} 1 & - \tan θ / 2 \\ \tan θ / 2 & 1 \end{matrix}]$
$= \cos^{2} \frac{θ}{2} [\begin{matrix} 1 - \tan^{2} θ / 2 & - 2 \tan θ / 2 \\ 2 \tan θ / 2 & 1 - \tan^{2} θ / 2 \end{matrix}]$
$= [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}]$

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