Expert: Paul Klarreich - 2/27/2007

Question
I have the basic 3 derivatives in my book but this one problem has a different answer in the back of the book. The problem: f(x)= cos(x)* sin(x). The back of the book says cos^2(x)- sin^2(x)=cos2x. How do I get that?

Answer
Questioner:   Christine
Category:  Calculus
 
Subject:  derivatives of trigonometric functions
Question:  I have the basic 3 derivatives in my book but this one problem has a different answer in the back of the book. The problem: f(x)= cos(x)* sin(x). The back of the book says cos^2(x)- sin^2(x)=cos2x. How do I get that?
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Hi, Christine,

Ah, the old 'back of the book' problem.  The exciting thing about trigonometry is the richness of the relationships and the ways we use them.  For example:

f(x) = cos x sin x = (1/2) sin(2x), by a well-known identity.  Then

f'(x) = (1/2)(2)cos(2x) = cos(2x)
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But you can do this by the product rule, too:

f'(x) = (cos x)(cos x) + (-sin x)(sin x)
f'(x) = cos^2(x) - sin^2(x)

Since two different ways to do the same example cannot give two different answers, these answers must be the same, even if they don't look the same.

In fact, it is easy to find the identity:

cos(2x) = cos^2(x) - sin^2(x)

in any standard listing of identities.  Look under double-angle formulas.  To prove it, you just use the 'sum' identity:

cos(A + B) = cos A cos B - sin A sin B

and put  A = x, B = x.

Just wait till you start integrating these things.  That's even more fun.