Question

Solve costheta [costheta sintheta
-sintheta costheta ] +sintheta [sintheta -costheta
costheta. sintheta]
​

Answer 1

$\textbf{To solve:}$

$\mathsf{cos\theta\left(\begin{array}{cc}cos\theta&sin\theta\\-sin\theta&cos\theta\end{array}\right)+sin\theta\left(\begin{array}{cc}sin\theta&-cos\theta\\cos\theta&sin\theta\end{array}\right)}$

$\textbf{Solution:}$

$\textsf{Consider,}$

$\mathsf{cos\theta\left(\begin{array}{cc}cos\theta&sin\theta\\-sin\theta&cos\theta\end{array}\right)+sin\theta\left(\begin{array}{cc}sin\theta&-cos\theta\\cos\theta&sin\theta\end{array}\right)}$

$=\mathsf{\left(\begin{array}{cc}cos^2\theta&cos\theta\;sin\theta\\-cos\theta\;sin\theta&cos^2\theta\end{array}\right)+\left(\begin{array}{cc}sin^2\theta&-cos\theta\;sin\theta\\cos\theta\;sin\theta&sin^2\theta\end{array}\right)}$

$=\mathsf{\left(\begin{array}{cc}cos^2\theta+sin^2\theta&cos\theta\;sin\theta-cos\theta\;sin\theta\\-cos\theta\;sin\theta+cos\theta\;sin\theta&cos^2\theta+sin^2\theta\end{array}\right)}$

$=\mathsf{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}$

$\therefore\mathsf{cos\theta\left(\begin{array}{cc}cos\theta&sin\theta\\-sin\theta&cos\theta\end{array}\right)+sin\theta\left(\begin{array}{cc}sin\theta&-cos\theta\\cos\theta&sin\theta\end{array}\right)=\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}$

Answer 2

ANSWER ⤵️

$\mathsf{cos\theta\left(\begin{array}{cc}cos\theta&sin\theta\\-sin\theta&cos\theta\end{array}\right)+sin\theta\left(\begin{array}{cc}sin\theta&-cos\theta\\cos\theta&sin\theta\end{array}\right)}$

$=\mathsf{\left(\begin{array}{cc}cos^2\theta&cos\theta\;sin\theta\\-cos\theta\;sin\theta&cos^2\theta\end{array}\right)+\left(\begin{array}{cc}sin^2\theta&-cos\theta\;sin\theta\\cos\theta\;sin\theta&sin^2\theta\end{array}\right)}$

$=\mathsf{\left(\begin{array}{cc}cos^2\theta+sin^2\theta&cos\theta\;sin\theta-cos\theta\;sin\theta\\-cos\theta\;sin\theta+cos\theta\;sin\theta&cos^2\theta+sin^2\theta\end{array}\right)}$

$=\mathsf{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}$

$\therefore\mathsf{cos\theta\left(\begin{array}{cc}cos\theta&sin\theta\\-sin\theta&cos\theta\end{array}\right)+sin\theta\left(\begin{array}{cc}sin\theta&-cos\theta\\cos\theta&sin\theta\end{array}\right)=\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}$