Question

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Answer 1

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Answer 2

$\large\underline{\sf{Solution-}}$

Given determinant is

$\rm :\longmapsto\:\rm \: \begin{gathered}\sf \left | \begin{array}{ccc}\cos(x + y)& - sin(x + y)&cos2y\\sinx&cosx& siny\\ - cosx& sinx& - cosy\end{array}\right | \end{gathered}$

We know that

$\boxed{ \rm{ cos(x + y) = cosxcosy - sinxsiny}}$

$\boxed{ \rm{ sin(x + y) = sinxcosy + sinycosx}}$

$\boxed{ \rm{ cos2x = {cos}^{2}x - {sin}^{2}x}}$

So, using these Identities, we get

$\rm \: = \: \begin{gathered}\sf \left | \begin{array}{ccc}\ \: cosxcosy - sinxsiny& - (sinxcosy + sinycosx)& {cos}^{2}y - {sin}^{2}y \\sinx&cosx& siny\\ - cosx& sinx& - cosy\end{array}\right | \end{gathered}$

can be rewritten as

$\rm \: = \: \begin{gathered}\sf \left | \begin{array}{ccc}\ \: cosxcosy - sinxsiny& - sinxcosy - sinycosx& {cos}^{2}y - {sin}^{2}y \\sinx&cosx& siny\\ - cosx& sinx& - cosy\end{array}\right | \end{gathered}$

$\boxed{ \bf{ \: OP \: R_1 \: \longmapsto \: R_1 \: + \: cosx \: R_3}}$

$\rm \: = \: \begin{gathered}\sf \left | \begin{array}{ccc}\ \: - sinxsiny& - sinycosx& - {sin}^{2}y \\sinx&cosx& siny\\ - cosx& sinx& - cosy\end{array}\right | \end{gathered}$

$\boxed{ \bf{ \: OP \: R_1 \: \longmapsto \: R_1 \: + \: siny \: R_2}}$

$\rm \: = \: \begin{gathered}\sf \left | \begin{array}{ccc}\ \: 0& 0& 0 \\sinx&cosx& siny\\ - cosx& sinx& - cosy\end{array}\right | \end{gathered}$

We know, If elements of any row or column are all 0, then determinant value is 0.

$\rm \:  =  \:  \: 0$

Hence, the value of

$\red{\rm :\longmapsto\:\rm \: \begin{gathered}\sf \left | \begin{array}{ccc}\cos(x + y)& - sin(x + y)&cos2y\\sinx&cosx& siny\\ - cosx& sinx& - cosy\end{array}\right | \end{gathered} = 0}$

Additional Information :-

1. The determinant value remains unaltered if rows and columns are interchanged.

2. The determinant value is 0, if two rows or columns are identical.

3. The determinant value is multiplied by - 1, if successive rows or columns are interchanged.

4. The determinant value remains unaltered if rows or columns are added or subtracted.