Prove that $\frac{{(cos (33^{\circ}))}^{2} - {(cos (57^{\circ}))}^{2}}{{(sin ({10.5}^{\circ}))}^{2} - {(sin ({34.5}^{\circ}))}^{2}} = - \sqrt{2}$ $((cos(33^@))^2-(cos(57^@))^2)/((sin(10.5^@))^2-(sin(34.5^@))^2)= -sqrt2$ ?

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Prove that $\frac{{(cos (33^{\circ}))}^{2} - {(cos (57^{\circ}))}^{2}}{{(sin ({10.5}^{\circ}))}^{2} - {(sin ({34.5}^{\circ}))}^{2}} = - \sqrt{2}$ $((cos(33^@))^2-(cos(57^@))^2)/((sin(10.5^@))^2-(sin(34.5^@))^2)= -sqrt2$ ?

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Answer 1 · 2017-10-22T13:51:07

Please see below.

Explanation:

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

(B) - ${cos}^{2} A - {sin}^{2} A = cos 2 A$ $cos^2A-sin^2A=cos2A$

(C) - $2 sin A cos A = sin 2 A$ $2sinAcosA=sin2A$ ,

(D) - $sin A + sin B = 2 sin (\frac{A + B}{2}) cos (\frac{A - B}{2})$ $sinA+sinB=2sin((A+B)/2)cos((A-B)/2)$ and

(E) - $sin A - sin B = 2 cos (\frac{A + B}{2}) sin (\frac{A - B}{2})$ $sinA-sinB=2cos((A+B)/2)sin((A-B)/2)$

$\frac{{cos}^{2} 33^{\circ} - {cos}^{2} 57^{\circ}}{{sin}^{2} {10.5}^{\circ} - {sin}^{2} {34.5}^{\circ}}$ $(cos^2 33^@-cos^2 57^@)/(sin^2 10.5^@-sin^2 34.5^@)$

= $\frac{{cos}^{2} 33^{\circ} - {sin}^{2} (90^{\circ} - 57^{\circ})}{({sin 10.5}^{\circ} + {sin 34.5}^{\circ}) ({sin 10.5}^{\circ} - {sin 34.5}^{\circ})}$ $(cos^2 33^@-sin^2 (90^@-57^@))/((sin10.5^@+sin34.5^@)(sin10.5^@-sin34.5^@))$ - used A

= $\frac{{cos}^{2} 33^{\circ} - {sin}^{2} 33^{\circ}}{- (2 {sin 22.5}^{\circ} {cos 12}^{\circ}) (2 {cos 22.5}^{\circ} {sin 12}^{\circ})}$ $(cos^2 33^@-sin^2 33^@)/(-(2sin22.5^@cos12^@)(2cos22.5^@sin12^@))$ - used D & E

= $\frac{{cos 66}^{\circ}}{- (2 {sin 22.5}^{\circ} {cos 22.5}^{\circ} \times 2 {sin 12}^{\circ} {cos 12}^{\circ})}$ $(cos66^@)/(-(2sin22.5^@cos22.5^@xx2sin12^@cos12^@)$ - used B

= $- \frac{sin (90^{\circ} - 66^{\circ})}{{sin 45}^{\circ} {sin 24}^{\circ}}$ $-(sin(90^@-66^@))/(sin45^@sin24^@)$ - used A & C

= $- \frac{{sin 24}^{\circ}}{\frac{1}{\sqrt{2}} {sin 24}^{\circ}}$ $-sin24^@/(1/sqrt2sin24^@)$

= $- \sqrt{2}$ $-sqrt2$

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Prove that $\frac{{(cos (33^{\circ}))}^{2} - {(cos (57^{\circ}))}^{2}}{{(sin ({10.5}^{\circ}))}^{2} - {(sin ({34.5}^{\circ}))}^{2}} = - \sqrt{2}$ $((cos(33^@))^2-(cos(57^@))^2)/((sin(10.5^@))^2-(sin(34.5^@))^2)= -sqrt2$ ?

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

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Prove that $\frac{{(cos (33^{\circ}))}^{2} - {(cos (57^{\circ}))}^{2}}{{(sin ({10.5}^{\circ}))}^{2} - {(sin ({34.5}^{\circ}))}^{2}} = - \sqrt{2}$ $((cos(33^@))^2-(cos(57^@))^2)/((sin(10.5^@))^2-(sin(34.5^@))^2)= -sqrt2$ ?