Compute the complex number $\cos \left(\frac\pi4 + \frac\pi4i\right)$

Question

Without a calculator or computer.

Hint: Consider expressions for $e^{\pm \theta i}$

i know that i cannot use a calculator or computer and that $\cos xy = \cos x\cos y - \sin x\sin y$ ive tried using $e^{\pm ix}$ is $\cos x\pm i\sin x$

this is supposed to be done without a calculator nor computer and must have a simple trick which im not sure about also we dont use use $\cosh$ or $\sinh$.

Answer 1

The simplest answer I have using the identities $$\cos (x+iy) = \cos x \cos iy - \sin x \sin iy$$ and $$\cos (ix) = \cosh x , \sin (ix) = i \sinh (x)$$ is $$\frac {\sqrt 2}{2} (\cosh \pi/4 + i \sinh \pi/4).$$

However, I can also use the identities $$\cosh (x) = \frac {e^x+e^{-x}}{2},\text { }\sinh (x) = \frac {e^x-e^{-x}}{2}$$ and get $$\frac {\sqrt 2}{2}(1+i)(e^{\pi/4}+e^{-\pi/4})$$ in terms of $e^x.$

Answer 2

Sometimes $\pm$ is not your friend.

The hint was to consider expressions (more than one expression) for $e^{\pm \theta i}$. There are two expressions you already know. If you write each of them individually rather than encoding both of them with $\pm$ as if they were one expression, you get this:

\begin{align} e^{ix} = \cos x + i\sin x, \\ e^{-ix} =\cos x - i\sin x. \end{align}

Now remember you are trying to evaluate $\cos x$ where $x$ is a known value.