Question

If \(z = i\log(2 - \sqrt{3})\), then the value of \(\cos z\) will be:

• A. \(i\)
• B. \(2i\)
• C. 1
• D. 2

Hint:

\(\cos(ix) = \cosh x\), \(\sin(ix) = i\sinh x\).

Answer 1

The Correct Option is D

Concept: \(\cos(i\theta) = \cosh \theta\).

Step 1: Write \(z\) in the form \(i\theta\). \[ z = i \log(2 - \sqrt{3}) \] Let \(\theta = \log(2 - \sqrt{3})\).

Step 2: Apply identity. \[ \cos z = \cos(i\theta) = \cosh \theta = \cosh(\log(2 - \sqrt{3})) \]

Step 3: Simplify using \(\cosh(\log t) = \frac{t + t^{-1}}{2}\). \[ \cosh(\log(2 - \sqrt{3})) = \frac{(2 - \sqrt{3}) + \frac{1}{2 - \sqrt{3}}}{2} \] Rationalize: \(\frac{1}{2 - \sqrt{3}} = 2 + \sqrt{3}\). So: \[ \frac{(2 - \sqrt{3}) + (2 + \sqrt{3})}{2} = \frac{4}{2} = 2 \]