Question

evaluate cos square 52 1/2 °-sin square 22 1/2°

Answer 1

here is your solution
sin square 22.5 can also be find by this method as a=22.5 and 2a=45

Answer 2

Answer:

$\cos^2 (52.5)-\sin^2 (22.5)=\frac{3-\sqrt3}{4\sqrt2}$

Step-by-step explanation:

Given : Expression $\cos^2 (52 \frac{1}{2}^\circ)-\sin^2 (22 \frac{1}{2}^\circ)$

To find : Evaluate the expression?

Solution :

We know, $\cos^2 A-\sin^2 B=\cos(A+B)\cos(A-B)$

$\cos^2 (52.5)-\sin^2 (22.5)=\cos(52.5+22.5)\cos(52.5-22.5)$

$\cos^2 (52.5)-\sin^2 (22.5)=\cos(75)\cos(30)$

$\cos^2 (52.5)-\sin^2 (22.5)=\frac{\sqrt3}{2}(\cos(45+30))$

Applying identity, $\cos(A+B)=\cos A\cos B-\sin A\sin B$

$\cos^2 (52.5)-\sin^2 (22.5)=\frac{\sqrt3}{2}(\cos 45\cos 30-\sin 45\sin 30)$

Substitute the values of the function,

$\cos^2 (52.5)-\sin^2 (22.5)=\frac{\sqrt3}{2}((\frac{1}{\sqrt2})(\frac{\sqrt3}{2})-(\frac{1}{\sqrt2})(\frac{1}{2}))$

$\cos^2 (52.5)-\sin^2 (22.5)=\frac{\sqrt3}{2}(\frac{\sqrt3}{2\sqrt2})-\frac{1}{2\sqrt2})$

$\cos^2 (52.5)-\sin^2 (22.5)=\frac{\sqrt3}{2}(\frac{\sqrt3-1}{2\sqrt2})$

$\cos^2 (52.5)-\sin^2 (22.5)=\frac{3-\sqrt3}{4\sqrt2}$

Therefore, $\cos^2 (52.5)-\sin^2 (22.5)=\frac{3-\sqrt3}{4\sqrt2}$