Question

Let $x$ be a real number and $i=\sqrt{-1}$. Then the real part of $\cos(ix)$ is

• A. $\sinh x$
• B. $\cosh x$
• C. $\cos x$
• D. $\sin x$

Hint:

Use Euler's relations: $\cos(ix)=\cosh x$ and $\sin(ix)=i\sinh x$. Complex arguments naturally convert trigonometric functions into hyperbolic ones.

Answer 1

The Correct Option is B

We use the Euler representation of cosine for complex arguments. The identity is: \[ \cos(z)=\frac{e^{iz}+e^{-iz}}{2}. \] Substitute $z=ix$: \[ \cos(ix)=\frac{e^{i(ix)}+e^{-i(ix)}}{2}=\frac{e^{-x}+e^{x}}{2}. \] This expression simplifies to the hyperbolic cosine function: \[ \cos(ix)=\cosh x. \] Since $\cosh x$ is purely real, the real part of $\cos(ix)$ is simply $\cosh x$.