Question

If $\left|\begin{array}{cc}{\sin }^2\alpha & {\cos }^2\alpha \\ {\cos }^2\alpha & {\sin }^2\alpha \end{array}\right|=0,\alpha ϵ(0,\pi )$ , 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

The correct option is C $\frac{\pi }{4}$ and $\frac{3\pi }{4}$

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

Thus the values of $\alpha$ are $\frac{\pi }{4}$ and $\frac{3\pi }{4}$.