1. ## Double angle formulas

Hi

I'm having some problems seeing how they've come to an answer in the course book, here we go.

$\displaystyle sin^2x \ cos^2x = \frac{1}{4}\ sin^2(2x)$

and then through several stages, some skipped, we get

$\displaystyle =\frac{1}{8}\ (1-cos(4x))$

Any help would be appreciated.

Thanks

2. We have $\displaystyle \sin 2x=2\sin x\cos x$ and $\displaystyle \cos 4x=1-2\sin^22x$

Now $\displaystyle \sin^2x\cos^2x=(\sin x\cos x)^2=\left(\frac{\sin 2x}{2}\right)^2=\frac{1}{4}\sin^22x=$

$\displaystyle =\frac{1}{8}\cdot 2\sin^22x=\frac{1}{8}(1-1+2\sin^22x)=\frac{1}{8}[1-(1-2\sin^22x)]=\frac{1}{8}(1-\cos 4x)$

3. look here

4. use cos2x formula

5. $\displaystyle cos(4x) = cos(2x + 2x) = .... = 8cos^4 x - 8cos^2 x + 1$

$\displaystyle cos^4 x = (1-sin^2 x)^2 = 1 - 2sin^2 x + sin^4 x$
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$\displaystyle sin^2 xcos^2 x = \frac{1}{8} (1 - cos(4x))$

$\displaystyle sin^2 xcos^2 x = \frac {1}{8} (1 - 8cos^4 x + 8cos^2 x - 1)$

$\displaystyle sin^2 xcos^2 x = -cos^4 x + cos^2 x$

$\displaystyle sin^2 xcos^2 x = -1 + 2sin^2x - sin^4 x + (1- sin^2 x)$

$\displaystyle sin^2 xcos^2 x = sin^2 x - sin^4 x$

$\displaystyle sin^2 xcos^2 x = sin^2 x (1 - sin^2 x)$

$\displaystyle cos^2 x = 1 - sin^2 x$

$\displaystyle cos^2 x = cos^2 x$

sorry for the long explanation :P

EDIT: just realized this wasn't really about proving the equalities.. meh I'll leave it unedited anyways )