Question

Simplify
$\cos \theta\left[\begin{array}{c c c} \cos \theta& \sin \theta \\ & & \\ - \sin \theta & \cos \theta &\end{array}\right] + \sin \theta\left[\begin{array}{c c c} \sin \theta& - \cos\theta \\ & & \\ \cos\theta & \sin\theta & \end{array}\right]$
​

Answer 1

Answer:

Cos theta + Sin theta

Step-by-step explanation

See answer in picture given by me.

Answer 2

Given Matrix :-

$\cos \theta\left[\begin{array}{c c c} \cos \theta& \sin \theta \\ & & \\ - \sin \theta & \cos \theta &\end{array}\right] + \sin \theta\left[\begin{array}{c c c} \sin \theta& - \cos\theta \\ & & \\ \cos\theta & \sin\theta & \end{array}\right]$

Solution :-

As $\sin\theta$ & $\cos\theta$ are out of matrix . So on , multiplying them with all elements of their respective matrixes , we have ;

${ : \implies \quad \left[\begin{array}{c c c} \cos² \theta& \sin \theta . \cos \theta \\ & & \\ - \sin \theta. \cos \theta & \cos² \theta &\end{array}\right] + \left[\begin{array}{c c c} \sin² \theta& - \cos\theta . \sin \theta \\ & & \\ \cos\theta .\sin\theta & \sin²\theta & \end{array}\right] }$

Now as the order of both matrixes are same i.e 2 × 2 , so we can add their corresponding elements . On adding their corresponding elements we have ;

${ : \implies \quad \left[\begin{array}{c c c} \cos² \theta + \sin²\theta & \sin \theta . \cos \theta - \cos\theta . \sin \theta \\ & & \\ - \sin \theta. \cos \theta + \sin\theta. \cos\theta & \cos² \theta + \sin²\theta &\end{array}\right] }$

We knows a Trigonometric identity i.e ;

$\quad \qquad { \bigstar { \underline { \boxed { \bf { \red { \cos²\theta + \sin²\theta = 1}}}}}}{\bigstar}$

Using this we have ;

${ : \implies \quad \left[\begin{array}{c c} 1 & \cancel{\sin \theta . \cos \theta} - \cancel{\cos\theta . \sin \theta} \\ & & \\ - \cancel{\sin \theta. \cos \theta} + \cancel{\sin\theta. \cos\theta} & 1 \end{array}\right] }$

${ : \implies \quad \left[\begin{array}{c c} 1 & 0 \\ & \\ 0 & 1 \end{array}\right]}$

${ \bigstar { \underline { \boxed { \bf { \red { \therefore\cos\theta\left[\begin{array}{c c c} \cos \theta& \sin \theta \\ & & \\ - \sin \theta & \cos \theta &\end{array}\right] + \sin \theta\left[\begin{array}{c c c} \sin \theta& - \cos\theta \\ & & \\ \cos\theta & \sin\theta & \end{array}\right] = \left[\begin{array}{c c} 1 & 0 \\ & \\ 0 & 1 \end{array}\right] }}}}}}{\bigstar}$