Question

If $\alpha +\beta ={90}^{\circ }$ and $\alpha =2\beta$, then ${\cos }^2\alpha +{\sin }^2\beta$ equal

• A. $1$
• B. $0$
• C. $\frac{1}{2}$
• D. $2$

Answer 1

The correct option is C $\frac{1}{2}$

It is given that
$\alpha =2\beta$
Hence
$\alpha +\beta ={90}^0$
Substituting, we get
$3\beta ={90}^0$
Hence
$\beta ={30}^0$
Therefore $\alpha ={60}^0$
Hence
${\sin }^2\beta +{\cos }^2\alpha$
$={\frac{1}{2}}^2+{\frac{1}{2}}^2$
$=\frac{1}{4}+\frac{1}{4}$
$=\frac{1}{2}$