Is it possible to compute this using Euler? $\frac{(\cos 1 + \cos 89)(\cos 2+\cos 88)\cdots(\cos 44 + \cos 46)}{\cos 1\cos 2\cdots\cos 44}$

Question

Is it possible to compute this using Euler?

$$\dfrac{(\cos(1) + \cos (89))(\cos(2)+\cos(88))...(\cos(44) + \cos(46))}{\cos(1)\cos(2)...\cos(44)}$$

I have easily computed this problem using trigonometric identities but I'm looking for a way where Euler is used. I used the fact that $\cos(x) = \dfrac{e^{ix} +e^{-ix}}{2}$. Surprisingly, there's no negative term.

Answer 1

Use http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html

$\cos x+\cos(90-x)=2\cos45\cos(45-x)$ for $1\le x\le44$

Alternatively

$\cos x+\cos(90-x)=\cos x+\sin x=\sqrt2\cos(45-x)$