Question

Find the value of $\cos \theta \left[ {\begin{array}{*{20}{c}}
  {\cos \theta }&{\sin \theta } \\
  { - \sin \theta }&{\cos \theta }
\end{array}} \right] + \sin \theta \left[ {\begin{array}{*{20}{c}}
  {\sin \theta }&{ - \cos \theta } \\
  {\cos \theta }&{\sin \theta }
\end{array}} \right] = $
A $\left[ {\begin{array}{*{20}{c}}
  0&0 \\
  0&0
\end{array}} \right]$
B $\left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&0
\end{array}} \right]$
C $\left[ {\begin{array}{*{20}{c}}
  0&1 \\
  1&0
\end{array}} \right]$
D $\left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right]$

Answer 1

Hint:First we will simplify the matrices then add the matrices. Then we will use the trigonometric identity ${\cos ^2}\theta + {\sin ^2}\theta = 1$ to solve the question and get the required solution.

Formula Used: ${\cos ^2}\theta + {\sin ^2}\theta = 1$

Complete step by step Solution:
 $\cos \theta \left[ {\begin{array}{*{20}{c}}
  {\cos \theta }&{\sin \theta } \\
  { - \sin \theta }&{\cos \theta }
\end{array}} \right] + \sin \theta \left[ {\begin{array}{*{20}{c}}
  {\sin \theta }&{ - \cos \theta } \\
  {\cos \theta }&{\sin \theta }
\end{array}} \right]$
We know that $a\left[ {\begin{array}{*{20}{c}}
  x&y \\
  z&u
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  {ax}&{ay} \\
  {az}&{au}
\end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}}
  {{{\cos }^2}\theta }&{\sin \theta \cos \theta } \\
  { - \sin \theta \cos \theta }&{{{\cos }^2}\theta }
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
  {{{\sin }^2}\theta }&{ - \sin \theta \cos \theta } \\
  {\sin \theta \cos \theta }&{{{\sin }^2}\theta }
\end{array}} \right]$
After adding, the matrices we will get
$ = \left[ {\begin{array}{*{20}{c}}
  {{{\sin }^2}\theta + {{\cos }^2}\theta }&{\sin \theta \cos \theta - \sin \theta \cos \theta } \\
  { - \sin \theta \cos \theta + \sin \theta \cos \theta }&{{{\cos }^2}\theta + {{\sin }^2}\theta }
\end{array}} \right]$
We know the identity ${\cos ^2}\theta + {\sin ^2}\theta = 1$
$ = \left[ {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right]$

Therefore, the correct option is (D).

Note:Students should do the calculations carefully to avoid any mistakes. And use the correct identities to solve the calculations correctly. And also keep in mind that ${\cos ^2}\theta + {\sin ^2}\theta = 1$.