Question

4sin Asin (A+π/3) sin(A+2π/3)=Sin3A​

Answer 1

To prove:- 4sinAsin (A+π/3) sin(A+2π/3)=3sinA

Proof:- Firstly let's solve LHS.

→2 Sin A [2 Sin (A+2 π/3) Sin(A+π/3)]

→ 2 Sin A [Cos (A+ 2 π/3 -A - π/3) Cos (A+ 2 Pπ/3 + A+π/3)] .... {Since we know Cos (A-B) Cos (A+B) = 2 Sin A sin B}

In this question for the values of A and B we have,

A=A+2π/3

A=A+2π/3B=A+π/3

So,

→2 sin A [(Cos π/3) {Cos (2 A + π)}]

→2 sin A [Cos π/3 (Cos 2A Cos πSin 2A Sin π)

→2 Sin A [Cos π/3 {Cos 2A (-1) Sin 2A (0)}]

→2 Sin A [Cos π/3 {-(cos2A)}]

→2 Sin A Cos π /3 +2 Sin A Cos 2A

→2 Sin A (1/2) + Sin (2A+ A) - Sin (2A A)

→Sin A + Sin 3A (SinA)

→Sin3A

Therefore,LHS=RHS

Hence proved!!

Remember two properties:-

☞Cos (A-B) Cos (A+B) = 2 Sin A sin B

☞2 Sin A Cos B = Sin (A+B) Sin (A-B)]