# Prove that ((cos(33^@))^2-(cos(57^@))^2)/((sin(10.5^@))^2-(sin(34.5^@))^2)= -sqrt2 ?

Oct 22, 2017

#### Explanation:

We use formulas (A) - $\cos A = \sin \left({90}^{\circ} - A\right)$,

(B) - ${\cos}^{2} A - {\sin}^{2} A = \cos 2 A$

(C) - $2 \sin A \cos A = \sin 2 A$,

(D) - $\sin A + \sin B = 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)$ and

(E) - $\sin A - \sin B = 2 \cos \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right)$

$\frac{{\cos}^{2} {33}^{\circ} - {\cos}^{2} {57}^{\circ}}{{\sin}^{2} {10.5}^{\circ} - {\sin}^{2} {34.5}^{\circ}}$

= $\frac{{\cos}^{2} {33}^{\circ} - {\sin}^{2} \left({90}^{\circ} - {57}^{\circ}\right)}{\left(\sin {10.5}^{\circ} + \sin {34.5}^{\circ}\right) \left(\sin {10.5}^{\circ} - \sin {34.5}^{\circ}\right)}$ - used A

= $\frac{{\cos}^{2} {33}^{\circ} - {\sin}^{2} {33}^{\circ}}{- \left(2 \sin {22.5}^{\circ} \cos {12}^{\circ}\right) \left(2 \cos {22.5}^{\circ} \sin {12}^{\circ}\right)}$ - used D & E

= (cos66^@)/(-(2sin22.5^@cos22.5^@xx2sin12^@cos12^@) - used B

= $- \frac{\sin \left({90}^{\circ} - {66}^{\circ}\right)}{\sin {45}^{\circ} \sin {24}^{\circ}}$ - used A & C

= $- \sin {24}^{\circ} / \left(\frac{1}{\sqrt{2}} \sin {24}^{\circ}\right)$

= $- \sqrt{2}$