How to simplify if $a > 0$ and $\cos(a) < 0$

Question

$$\sqrt{\cos (a)} \sinh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia}\right)}\right)+\sqrt{\cos (a)} \cosh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia }\right)}\right)=$$

Answer 1

Hint 1: $\frac12 (e^{ia}+e^{-ia})=\cos a$

Hint 2: $\cosh x+\sinh x=e^x$

Hint 3: $e^{\log x}=x$

Answer 2

With $\log(a)$ as the natural logarithm $\ln(a)$ gives us:

$$\sqrt{\cos (a)} \sinh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia}\right)}\right)+\sqrt{\cos (a)} \cosh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia }\right)}\right)=$$

$$\sqrt{\cos (a)} \sinh \left(\ln (2) a^{\cos(a)}\right)+\sqrt{\cos (a)} \cosh \left(\ln (2) a^{\cos(a)}\right)=$$

Evaluates to:

$$2^{a^{\cos(a)}}\sqrt{\cos(a)}$$