Question

The value of sin 3A sin³ A + cos 3A cos³ A is ______.

पर्याय

• cos³ 2A

• cos² A

• sin³ A

• sin³ 2A

Answer 1

The value of sin 3A sin³ A + cos 3A cos³ A is cos³ 2A.

Explanation:

We know that,

sin 3A = 3 sin A – 4 sin³ A

`\implies` sin³ A = `(3 sin A - sin 3A)/4`

Similarly, cos 3A = 4 cos³ A – 3 cos A

`\implies` cos³ A = `(cos 3A + 3 cos A)/4`

∴ LHS = sin 3A sin³ A + cos 3A cos³ A

`\implies` LHS = `sin 3A{(3 sin A - sin 3A)/4} + cos 3A{(cos 3A + 3 cos A)/4}`

`\implies` LHS = `1/4{3(cos A cos 3A + sin A sin 3A) + (cos^2 3A - sin^2 3A)}`

`\implies` LHS = `1/4{3 cos(3A - A) + cos 2(3A)}`

`\implies` LHS = `1/4{3 cos 2A + cos 3(2A)}`

`\implies` LHS = `1/4{3 cos 2A + (4 cos^3 2A - 3 cos 2A)}`

= cos³ 2A

= RHS