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Sine Rule

If A + B + ...

Question

If $A + B + C = \frac{π}{2}$ , prove the following
$s i n 2 A + s i n 2 B - s i n 2 C = 4 c o s A c o s B s i n C$ .

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Solution

$\begin{matrix} W e h a v e A + B + C = π ∴ A i s s u p p l e m e n t a n g l e o f B + C s i m i l a r i l y B a n d C t o o t h e r s N o w, sin 2 A + sin 2 B - sin 2 C = 2 sin A cos A + 2 sin B cos B - 2 sin C cos C = 2 sin (B + C) cos A + 2 sin (A + C) cos B - 2 sin (A + B) cos C = (2 sin B s i n C + 2 c o s B s i n C) cos A + (2 sin A cos C + 2 cos A sin C) cos B - (2 S i n A cos B + 2 cos A sin B) cos C = 4 cos B sin C cos A \end{matrix}$
Proved.

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Similar questions

A + B + C = π

, then prove the following

sin 2 A + sin 2 B + sin 2 C = 4 sin A . sin B . sin C

A + B + C = π

, then prove that

{sin}^{2} A + {sin}^{2} B + {sin}^{2} C = 2 + 2 cos A \cdot cos B \cdot cos C

sin 2 A + sin 2 B - sin 2 C = 4 cos A cos B sin C

A + B + C = π

sin 2 A + sin 2 B + sin 2 C = 4 sin A sin B sin C

A + B + C = π

then the expression

\frac{sin 2 A + sin 2 B - sin 2 C}{sin 2 A + sin 2 B + sin 2 C}

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