Differentiating the identity to develop another identity

[meeklobraca]

Homework Statement

Differentiate the identity sin2x = 2sinxcosx to develop the identity for cos2x, in terms on sin x and cos x

Homework Equations

The Attempt at a Solution

Im not sure where to start with this one. Should I find the derivative of both sides of the equation, and then where do I go from there? The right side of that equation is 2cos^2x + sin^2x but I am not sure how I can use that.

Your help is appreciated as always!

 

[shaggymoods]

I believe you differentiated the right-hand side incorrectly:

$\frac{d}{dx}(2\sin(x)\cos(x)) = 2(\cos^{2}(x) + (-\sin^{2}(x)))$ $\Rightarrow 2\cos(2x) = 2(\cos^{2}(x) - \sin^{2}(x))$$\Rightarrow \cos(2x) = \cos^{2}(x) - \sin^{2}(x)$which is a well-known double-angle formula.

 

[praharmitra]

Actually, once you find the dertivative of both sides, you will get the identity instantly(barring some cancellations).

Find the derivative of the left and right side (what u have written is not correct), using product rule, and see what cancels, on both sides

 

[meeklobraca]

Thank you very much. I mistakenly used part of the quotient rule instead of the product rule, meaning I subtracted instead add added which caused me to get the + sin ^2x

Cheers guys!