Question

If z = $i\log (2-\sqrt{3})$, then value of cos z is ______ [use $e^{\left({\log }_{e}A\right)}=A$]

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Answer 1

Z = $i\log (2-\sqrt{3})$

$e^{iz}=e^{i\left[i\log \right(2-\sqrt{3}\left)\right]}$

= $e^{-log(2-\sqrt{3})}=\frac{1}{2-\sqrt{3}}$

$e^{-iz}=e^{-i\left(ilog\right(2-\sqrt{3}\left)\right)}=2-\sqrt{3}$

Cos z = $\frac{e^{iz}+e^{-iz}}{2}$

= $[\frac{1}{2-\sqrt{3}}+2-\sqrt{3}]\frac{1}{2}$

= $[2+\sqrt{3}+2-\sqrt{3}]\frac{1}{2}$ $[\because \frac{1}{2-\sqrt{3}}=2+\sqrt{3}]$

= 2