Trouble computing the cosine of a complex number

[gex]

Mentor note: Thread moved from technical section, so missing the homework template.
Hi all, I have a homework problem which asks me to compute the complex number cos(π/4 + π/4 i).
I've been playing around with it for a while now and just can't seem to get the answer I get when using Wolfram Alpha to verify. Attached is my most recent attempt at solving the problem, I'd love some input as to where I'm going wrong. Thank you in advance.

 

[Mark44]

Hi all, I have a homework problem which asks me to compute the complex number cos(π/4 + π/4 i).
I've been playing around with it for a while now and just can't seem to get the answer I get when using Wolfram Alpha to verify. Attached is my most recent attempt at solving the problem, I'd love some input as to where I'm going wrong. Thank you in advance.View attachment 224899

You have a sign error. In your "therefore" line (the line with ∴), in the middle expression, the part in parentheses should be $(\frac \pi 4 + i\frac \pi 4)$. You have a minus where it should be a plus.

 

[FactChecker]

In the second line, factor out $e^{-\frac{\pi}{4}}$ and apply the $2cos(\theta)$ identity to the other factor.

 

[Merlin3189]

The last expression on the line starting, " ∴ cos( " is wrong -i(n-ni) = -ni +n

 

[gex]

Thank you so much everyone, I feel foolish for my careless sign error. Most of all I feel relieved that my approach wasn't flawed. I get the correct answer now :)