Question

The value of the determinant $\left|\begin{array}{ccc}\cos \left(x+y\right)& -\sin \left(x+y\right)& \cos 2y\\ \sin x& \cos x& \sin y\\ -\cos x& \sin x& \sin y\end{array}\right|$ depends on __________ only.

Answer 1

Let ∆ = $\left|\begin{array}{ccc}\cos \left(x+y\right)& -\sin \left(x+y\right)& \cos 2y\\ \sin x& \cos x& \sin y\\ -\cos x& \sin x& \sin y\end{array}\right|$



$$\begin{gathered} ∆=\left|\begin{array}{ccc}\cos \left(x+y\right)& -\sin \left(x+y\right)& \cos 2y\\ \sin x& \cos x& \sin y\\ -\cos x& \sin x& \sin y\end{array}\right| \\ \mathrm{Applying}{R}_2\to \sin y{R}_{2}and{R}_3\to \cos y{R}_3 \\ =\frac{1}{\sin y\cos y}\left|\begin{array}{ccc}\cos \left(x+y\right)& -\sin \left(x+y\right)& \cos 2y\\ \sin x\sin y& \cos x\sin y& {\sin }^{2}y\\ -\cos x\cos y& \sin x\cos y& \sin y\cos y\end{array}\right| \\ \mathrm{Applying}{R}_1\to R_1+R_2+R_3 \\ =\frac{1}{\sin y\cos y}\left|\begin{array}{ccc}\cos \left(x+y\right)+\sin x\sin y-\cos x\cos y& -\sin \left(x+y\right)+\cos x\sin y+\sin x\cos y& \cos 2y+{\sin }^{2}y+\sin y\cos y\\ \sin x\sin y& \cos x\sin y& {\sin }^{2}y\\ -\cos x\cos y& \sin x\cos y& \sin y\cos y\end{array}\right| \\ =\frac{1}{\sin y\cos y}\left|\begin{array}{ccc}\cos \left(x+y\right)-\cos \left(x+y\right)& -\sin \left(x+y\right)+\sin \left(x+y\right)& \cos 2y+{\sin }^{2}y+\sin y\cos y\\ \sin x\sin y& \cos x\sin y& {\sin }^{2}y\\ -\cos x\cos y& \sin x\cos y& \sin y\cos y\end{array}\right| \\ =\frac{1}{\sin y\cos y}\left|\begin{array}{ccc}0& 0& \cos 2y+{\sin }^{2}y+\sin y\cos y\\ \sin x\sin y& \cos x\sin y& {\sin }^{2}y\\ -\cos x\cos y& \sin x\cos y& \sin y\cos y\end{array}\right| \\ \mathrm{Expanding}\mathrm{through}{R}_1 \\ =\frac{1}{\sin y\cos y}\left[\left(\cos 2y+{\sin }^{2}y+\sin y\cos y\right)\left({\sin }^{2}x\sin y\cos y+{\cos }^{2}x\sin y\cos y\right)\right] \\ =\frac{1}{\sin y\cos y}\left[\left(\cos 2y+{\sin }^{2}y+\sin y\cos y\right)\sin y\cos y\left({\sin }^{2}x+{\cos }^{2}x\right)\right] \\ =\left[\left(\cos 2y+{\sin }^{2}y+\sin y\cos y\right)\left({\sin }^{2}x+{\cos }^{2}x\right)\right] \\ =\left[\left(\cos 2y+{\sin }^{2}y+\sin y\cos y\right)\left(1\right)\right] \\ =\left(\cos 2y+{\sin }^{2}y+\sin y\cos y\right) \\ =\left(1-2{\sin }^{2}y+{\sin }^{2}y+\sin y\cos y\right) \\ =\left(1-{\sin }^{2}y+\sin y\cos y\right) \end{gathered}$$

Hence, the value of the determinant $\left|\begin{array}{ccc}\cos \left(x+y\right)& -\sin \left(x+y\right)& \cos 2y\\ \sin x& \cos x& \sin y\\ -\cos x& \sin x& \sin y\end{array}\right|$ depends on y only.