Question

If $\begin{bmatrix}\sin ^{2} \alpha \cos ^{2} \alpha \\ \cos ^{2} \alpha \sin ^{2} \alpha\end{bmatrix}=0, \alpha \in(0, \pi)$, then the values of $\alpha$ are

• A. $\frac{\pi}{2}$ and $\frac{\pi}{12}$
• B. $\frac{\pi}{2}$ and $\frac{\pi}{6}$
• C. $\frac{\pi}{4}$ and $\frac{3\pi}{4}$
• D. $\frac{\pi}{6}$ and $\frac{\pi}{3}$
• E. $\frac{\pi}{2}$ and $\frac{\pi}{3}$

Answer 1

Answer: (C)

Given, $\begin{bmatrix}\sin ^{2} \alpha & \cos ^{2} \alpha \\ \cos ^{2} \alpha & \sin ^{2} \alpha\end{bmatrix}=0$
$\Rightarrow \sin ^{4} \alpha-\cos ^{4} \alpha=0$
$\Rightarrow \left(\sin ^{2} \alpha-\cos ^{2} \alpha\right)\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)=0$
$\Rightarrow -\cos 2 \alpha(1)=0$
$\Rightarrow \cos 2 \alpha=0$
$\Rightarrow 2 \alpha=\frac{\pi}{2}$ and $\frac{3 \pi}{2}$
$\Rightarrow \alpha=\frac{\pi}{4}$ and $\frac{3 \pi}{4}$