Discussion
Let \(x\) be a real number and \(i = \sqrt{-1}\). Then the real part of \(\cos(ix)\) is
Answer: B
The real part of cos(ix) is cosh x.
From the exponential form of the cosine:
\[ \begin{aligned} \cos z &= \frac{e^{iz} + e^{-iz}}{2} \\ \cos(ix) &= \frac{e^{-x} + e^{x}}{2} = \cosh x \end{aligned} \]
For real x this is already real, so the real part is cosh x itself and the imaginary part is zero. sinh x is the corresponding result for \(-i\sin(ix)\), and cos x and sin x would be the answers only if the argument were real.
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