Question

Show that cos(3x) = cos3(x) − 3 sin2(x)cos(x). (Hint: Use cos(2x) = cos2(x) − sin2(x) and the cosine sum

formula.)

Answer 1

Explanation:

To prove:

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

Identities used:

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

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

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

Taking the LHS:

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

Using identity 1:

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

Using identities 2 and 3:

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

As, LHS = RHS

Hence proved