Question

If $z=i\log (2-\sqrt{3})$ then the value of $\cos z$ is

Answer 1

Given that:-

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

To find:- $\cos z=?$

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

Multiplying both sides by $i$, we have

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

$iz=-\log (2-\sqrt{3})(\because i^2=-1)$

$\Rightarrow -iz=\log (2-\sqrt{3})$

$e^{-iz}=2-\sqrt{3}$

As we know that,

$\cos \theta =\frac{e^{i\theta }+e^{-i\theta }}{2}$

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

$\Rightarrow \cos z=\frac{\left(\frac{1}{e^{-iz}}\right)+e^{-iz}}{2}$

$\Rightarrow \cos z=\frac{1+{\left(e^{-iz}\right)}^2}{2e^{-iz}}$

Substituting the value of $e^{-iz}$ in the above equation, we have

$\therefore \cos z=\frac{1+{(2-\sqrt{3})}^2}{2(2-\sqrt{3})}$

$\Rightarrow \cos z=\frac{1+(2^2+{\sqrt{3}}^2-2\times 2\times \sqrt{3})}{2(2-\sqrt{3})}$

$\Rightarrow \cos z=\frac{1+4+3-4\sqrt{3}}{2(2-\sqrt{3})}$

$\Rightarrow \cos z=\frac{8-4\sqrt{3}}{(4-2\sqrt{3})}$

$\Rightarrow \cos z=\frac{2(4-2\sqrt{3})}{(4-2\sqrt{3})}$

$\Rightarrow \cos z=2$