Double-Angle Formula: Solving for (1+\cosh(v))(\cosh(v)-\cos(u))

Ted123 · 2011-11-30T18:53:28+00:00

I need to get from \cosh^2(v)-2\cos(u)\cosh(v) + 4\cos^2(\frac{u}{2})\sinh^2(\frac{v}{2}) + 1 to (1+\cosh(v))(\cosh(v)-\cos(u)) using double angle formulae.

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

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