If the product of matrices $\begin{bmatrix}cos^2θ& cos θsinθ\\cosθsinθ & sin^2θ\end{bmatrix} $ and $\begin{bmatrix}cos^2\beta & cos\beta sin \beta \\cos \beta sin\beta & sin^2\beta \end{bmatrix}$ is a null matrix, then θ and $\beta $ differ by

an odd integral multiple of $\frac{\pi}{2}$

The correct answer is Option (4) → an odd integral multiple of $\frac{\pi}{2}$

$\begin{bmatrix}\cos^2θ& \cos θ\sinθ\\\cos θ\sin θ & \sin^2θ\end{bmatrix}\begin{bmatrix}\cos^2\beta & \cos\beta \sin \beta \\\cos \beta \sin\beta & \sin^2\beta \end{bmatrix}=0$

$\begin{bmatrix}\cos θ\cos β(\cos θ\cos β+\sin θ\sin β)&\cos θ\sin β(\cos θ\cos β+\sin θ\sin β)\\\sin θ\cos β(\cos θ\cos β+\sin θ\sin β)&\sin θ\sin β(\cos θ\cos β+\sin θ\sin β)\end{bmatrix}$

$=\begin{bmatrix}\cos θ\cos β\cos(θ-β)&\cos θ\sin β\cos(θ-β)\\\sin θ\cos β\cos(θ-β)&\sin θ\sin β\cos(θ-β)\end{bmatrix}$

so $θ-β$ is an odd multiple of $\frac{\pi}{2}$