Help with this math Transformation

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Can anyone explain why (cos(x))^2 = (1+cos(2x))/2 ?
 
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cos(x)=(Exp(ix)+Exp(-ix))/2 right? do u know this formula?
square before and after the '='
then simplify it..
 
If you don't want to use complex numbers, remember that cos2x= cos2 x- sin2 x so that the right hand side is (1+ cos2 x- sin2x)/2. Do you see an "obvious" way to simplify 1+ cos2 x- sin2x?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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