Question

Differentiate the identity sin2x = 2sinxcosx to develop the identity for cos2x, in terms f sinx and cosx. Help!?

Answer 1

`cos2x = cos^2x-sin^2x`

Explanation:

We know that the identity sought is:

`cos2x = cos^2x-sin^2x`

as this comes from the cosine sum of angles formula.

However, the intention is to derive this identity from finding the derivative of the sine sum of angles formula.

We are given that:

`sin2x -= 2sinxcosx`

And so differentiating both sides wrt `x` by applying the product rule we get:

`2cos2x -= (2sinx)(-sinx) + (2cosx)(cosx)`

`:. cos2x -= -sin^2x + cos^2x`
`:. cos2x -= cos^2x -sin^2x` QED