Initial program 43.5
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
Taylor expanded around 0 0.8
\[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2} \cdot \sin y i\right))\]
Simplified0.8
\[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{(\left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right) \cdot x + \left({x}^{5} \cdot \frac{1}{60}\right))_*}}{2} \cdot \sin y i\right))\]
Taylor expanded around inf 0.8
\[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \color{blue}{\left(\frac{1}{6} \cdot \left({x}^{3} \cdot \sin y\right) + \left(x \cdot \sin y + \frac{1}{120} \cdot \left({x}^{5} \cdot \sin y\right)\right)\right)} i\right))\]
Simplified0.8
\[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \color{blue}{(x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x\right)\right) + \left((\frac{1}{120} \cdot \left({x}^{5}\right) + x)_*\right))_* \cdot \sin y} i\right))\]
Final simplification0.8
\[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \sin y \cdot (x \cdot \left(\left(x \cdot \frac{1}{6}\right) \cdot x\right) + \left((\frac{1}{120} \cdot \left({x}^{5}\right) + x)_*\right))_* i\right))\]
