Can This Exponential Identity Be Proven?

hariyo
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Does anyone know how to prove this?

exp(jAt)+exp(jBt)= 2exp(j(A+B)/2)cos((A-B)/2)
 
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Sorry I think i know by now. Here is the solution.

exp(jAt)+exp(jBt)= 2exp(j(A+B)/2)cos((A-B)/2)

(exp(jAt)+exp(jBt) ) exp(-j(A+B)t/2) exp(+j(A+B)t/2)

(exp(j(A-B)t/2)+exp(-j(A-B)t/2) ) exp(+j(A+B)t/2)

cos((A-B)t/2) 2exp(j(A+B)/2)t
 

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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