# Product of Complex Numbers with cosh and sin

**Calculus**Level 5

\[\prod_{k=1}^\infty \frac{\cosh\left(k^2+k+\frac{1}{2}\right) + i \sin\left(k+\frac{1}{2}\right)}{\cosh\left(k^2+k+\frac{1}{2}\right) - i \sin\left(k+\frac{1}{2}\right)}.\] Evaluate the product above. The answer should be of the form \[\frac{a_2 e^2 + a_1 e + a_0 + i \left( b_2 e^2 + b_1 e + b_0\right)}{c_2 e^2 + c_1 e + c_0}.\] Enter \(2^2 a_2 + 2^1 a_1 + 2^0 a_0 + 3^2 b_2 + 3^1 b_1 + 3^0 b_0 + 5^2 c_2 + 5^1 c_1 + 5^0 c_0\) for the answer.