If A+B+C=π,...

Question

If $A + B + C = π$ , prove that $(sin A + sin B + sin C) (- sin A + sin B + sin C) (sin A - sin B + sin C) (sin A + sin B - sin C) = 4 {sin}^{2} A {sin}^{2} B {sin}^{2} C$ .

Open in App

Solution

Then L.H.S. $= [4 cos \frac{A}{2} cos \frac{B}{2} cos \frac{C}{2}] [4 sin \frac{B}{2} sin \frac{C}{2} cos \frac{A}{2}]$ $\cdot [4 sin \frac{C}{2} sin \frac{A}{2} cos \frac{B}{2}] [4 sin \frac{A}{2} sin \frac{B}{2} cos \frac{C}{2}]$
$= 4 \times 64 ({sin}^{2} \frac{A}{2} {cos}^{2} \frac{A}{2})$ $() ()$
$= 4 {(2 sin \frac{A}{2} cos \frac{A}{2})}^{2}$ $() ()$
$= 4 {sin}^{2} A {sin}^{2} B {sin}^{2} C$ .

Suggest Corrections

Similar questions

\frac{sin (A - B)}{sin A sin B} + \frac{sin (B - C)}{sin B sin C} + \frac{sin (C - A)}{sin C sin A} = 0

A B C

(sin A + sin B + sin C) (sin A + sin B - sin C) = 3 sin A sin B

△ A B C,

(sin A + sin B + sin C) (sin A + sin B - sin C) = 3 sin A sin B,

C

\sum \frac{sin (C - A)}{sin C sin A}

sin A, sin B, sin C

Δ

A B C

A, B, C

a, b, c

(sin A + sin B + sin C) (sin A + sin C - sin B) = μ sin A sin C

sin A = a k, sin B = b k, sin C = c k

μ