Deriving Lagrange's Trig Identity: Real Part of Complex # in Exp Form

Lchan1
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Homework Statement


I did the question with help, but did not understand why did we multiply e^-i(x/2)
How do I know what to multiply for getting the real part of a complex number in exponential form?


Homework Equations





The Attempt at a Solution

 
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Lchan1 said:

Homework Statement


I did the question with help, but did not understand why did we multiply e^-i(x/2)
How do I know what to multiply for getting the real part of a complex number in exponential form?


Homework Equations





The Attempt at a Solution

You need to give us more information. I don't know what problem you're trying to solve.
 


Show us what you're doing and where you're getting stuck.
 
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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