Question

Evaluate: ${\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos (x+\frac{\pi }{2})\cos x$

Answer 1

Given, ${\sin }^{2}x+{\sin }^2(x+\frac{\pi }{3})+\cos (x+\frac{\pi }{2})\cos x$

$={\sin }^{2}x+{(\sin x\cos \frac{\pi }{3}+\cos x\sin \frac{\pi }{3})}^2-\sin x\cos x$

$={\sin }^{2}x+{(\sin x\frac{1}{2}+\cos x\frac{\sqrt{3}}{2})}^2-\sin x\cos x$

$={\sin }^{2}x+\frac{1}{4}{(\sin x+\sqrt{3}\cos x)}^2-\sin x\cos x$

$={\sin }^{2}x+\frac{1}{4}({\sin }^{2}x+3{\cos }^{2}x+2\sqrt{3}\sin x\cos x)-\sin x\cos x$

$=\frac{5}{4}{\sin }^{2}x+\frac{3}{4}{\cos }^{2}x+\frac{\sqrt{3}}{2}\sin x\cos x-\sin x\cos x$

$=\frac{5}{4}{\sin }^{2}x+\frac{3}{4}{\cos }^{2}x+(\frac{\sqrt{3}}{2}-1)\sin x\cos x$

$\therefore f\left(x\right)=\frac{1}{4}(5{\sin }^{2}x+3{\cos }^{2}x)+(\frac{\sqrt{3}}{2}-1)\sin x\cos x$