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QUESTION Using the usual expansion for sin(A+B)(how would you
prove this for A,B complex?)prove that if $z=x+iy$ then $$\sin
(x+iy)=\sin x\cosh y+i\cos x\sinh y.$$


ANSWER Usual expansion for sin(a+b) is $$\sin(a+b)=\sin A\cos
B+\sin B\cos A$$ (This may be proved for complex $A,B$ by
expressing both sides in terms of the exponential function.)Now
$\cos iy=\cosh y, \sin iy= i\sinh y$, giving the result.

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