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expanding sin(x+iy) and equating real and imaginary parts, we get:

##\cos(\alpha)=\sin(x)\cosh(y)## -- (1)

and ## \sin(\alpha)=\cos(x)\sinh(y) ## -- (2)

if we use ##\cos^2(\alpha)+\sin^2(\alpha)=1 ## and put the values from (1) and (2) , we can get the required expression. My question is that, is there an alternative to do this? Can I prove it by directly using the values of cosh2y and cos2x [from (1) and (2)]? I tried but it seems impossible. Why is it so?