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  4. The value of the determinant |sin2(x+(3π/2))&sin 2(x+(5π/2))&sin2 (x+(7π/2)) sin(x+(3π/2))&sin(x+(5π/2))&sin(x+(7π/2)) sin(x-(3π/2))&sin(x-(5π/2))&sin(x+(7π/2))| is

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Q. The value of the determinant
$\begin{vmatrix}sin^{2}\left(x+\frac{3\pi}{2}\right)&sin ^{2}\left(x+\frac{5\pi}{2}\right)&sin^{2} \left(x+\frac{7\pi}{2}\right)\\ sin\left(x+\frac{3\pi}{2}\right)&sin\left(x+\frac{5\pi}{2}\right)&sin\left(x+\frac{7\pi}{2}\right)\\ sin\left(x-\frac{3\pi}{2}\right)&sin\left(x-\frac{5\pi}{2}\right)&sin\left(x+\frac{7\pi}{2}\right)\end{vmatrix}$ is

Determinants

Solution:

Apply $C_{3} \rightarrow C_{3}-C_{1}$
$\begin{vmatrix}sin^{2}\left(x+\frac{3\pi}{2}\right)&sin ^{2}\left(x+\frac{5\pi}{2}\right)&sin\left(2x+5\pi\right)sin\left(2\pi\right)\\ sin\left(x+\frac{3\pi}{2}\right)&sin\left(x+\frac{5\pi}{2}\right)&2cos\left(x+\frac{5\pi}{2}\right)sin\left(\pi\right)\\ sin\left(x-\frac{3\pi}{2}\right)&sin\left(x-\frac{5\pi}{2}\right)&2\,cos\left(x-\frac{5\pi}{2}\right)sin\left(-\pi\right)\end{vmatrix}=0$
$\therefore $ All elements of $C_{3}$ are zero