Question

Show that sin(3x) = 3sin(x)cos2(x) − sin3(x). (Hint: Use sin(2x) = 2sin(x)cos(x) and the sine sum formula.)

Answer 1

Explanation:

To prove:

`\sin 3x=3\sin x\cos^2 x-\sin^3x`

Identities used:

`\sin(A+B)=\sin A\cos B+\cos A\sin B` ......(1)

`\sin 2A=2\sin A\cos A` ........(2)

`\cos 2A=\cos^2A-\sin^2 A` .......(3)

Taking the LHS:

`\Rightarrow \sin 3x=\sin (x+2x)`

Using identity 1:

`\Rightarrow \sin (x+2x)=\sin x\cos 2x+\cos x\sin 2x`

Using identities 2 and 3:

`\Rightarrow \sin x(\cos ^2x-\sin^2 x)+\cos x(2\sin x\cos x)\\\\\Rightarrow \sin x\cos^2x-\sin^3x+2\sin x\cos^2x\\\\\Rightarrow 3\sin x\cos^2x-\sin^3x`

As, LHS = RHS

Hence proved