Question

The value of ${\sin }^{2}52{\frac{1}{2}}^{\circ }-{\sin }^{2}22{\frac{1}{2}}^{\circ }$ is

• A. $\frac{\sqrt{3}-1}{2\sqrt{2}}$
• B. $\frac{\sqrt{3}+1}{2\sqrt{2}}$
• C. $\frac{\sqrt{3}-1}{4\sqrt{2}}$
• D. $\frac{\sqrt{3}+1}{4\sqrt{2}}$

Answer 1

The correct option is D $\frac{\sqrt{3}+1}{4\sqrt{2}}$

We know that,
${\sin }^{2}A-{\sin }^{2}B=\sin (A+B)\cdot \sin (A-B)$
${\sin }^{2}52{\frac{1}{2}}^{\circ }-{\sin }^{2}22{\frac{1}{2}}^{\circ }=\sin {75}^{\circ }\sin {30}^{\circ }=\sin {30}^{\circ }\sin (45+30{)}^{\circ }=\sin 30°(\sin 45°\cos 30°+\sin 30°\cos 45°)=\frac{1}{2}\times \frac{\sqrt{3}+1}{2\sqrt{2}}=\frac{\sqrt{3}+1}{4\sqrt{2}}$