Question

If $2\cos 2A=\sqrt{3}$ and $A$ is acute, find:

${\sin }^2({75}^0-A)+{\cos }^2({45}^0+A)$

Answer 1

Given: $2\cos 2A=\sqrt{3}$
$\cos 2A=\frac{\sqrt{3}}{2}$
$\cos 2A=\cos 30$
$2A=30$
$A={15}^{\circ }$
Now, ${\sin }^2(75-A)+{\cos }^2(45+A)={\sin }^{2}60+{\cos }^{2}60=1$ (Since, ${\sin }^2\theta +{\cos }^2\theta =1$)