Establish the identity. `cos^2(2u)-sin^2(2u)=cos(4u)`

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lemjay eNotes educator| Certified Educator

`cos^2(2u)-sin^2(2u) = cos (4u)`

To prove, simplify left side. In simplifying, let `theta=2u` .

`cos^2theta-sin^2theta=cos (4u)`

Then, apply the double angle identity of cosine which is `cos^2(2theta)= cos^2theta - sin^2 theta` .

`cos(2theta) = cos (4u)`

Then, substitute back  `theta = 2u` .

`cos (2*2u)=cos(4u)`

`cos(4u) = cos(4u)`

Hence, `cos^2(2u)-sin^2(2u) = cos (4u)` is an identity.

steveschoen eNotes educator| Certified Educator

Hi, spgrmd,

Let's try this.  The formula for the sum of angles is:

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

If A and B = 2u, then we would have:

cos(2u + 2u) = cos 2u cos 2u - sin 2u sin 2u


cos(4u) = cos^2 (2u) - sin^2 (2u)

I hope this helps, spgrmd.  Good luck.