Better:
The question
Express tan(1 - i) in the form a + ib, for a and b real
My attempt
I used the fact that:
sin(z) = sin(x)cosh(y) + icos(x)sinh(y)
cos(z) = cos(x)cosh(y) - isin(x)cosh(y)
I substituded these into tan(z) = \frac{sin(z)}{cos(z)}, and after quite a bit of algebra, got the following:
I tried simplifying further, but I got a mess involving coth(1), cot(1) and other assorted trig functions. The solution in my text is:
Could someone guide me on how they got to that solution? Thank you.
I don't know that it has a particular name, it's just the addition formula for tan. Incidentally, any such formula that hold for real numbers also holds for complex numbers, by the principle of permanence of functional equations.