Question

Solve $\sqrt{\frac{1+\cos {30}^0}{1-\cos {30}^0}}$

Answer 1

Solution:-

As we know that,

$\cos 2x=1-2{\sin }^{2}x\Rightarrow 1-\cos 2x=2{\sin }^{2}x$

$\cos 2x=2{\cos }^{2}x-1\Rightarrow 1+\cos 2x=2{\cos }^{2}x$

$\therefore \sqrt{\frac{1+\cos 30}{1-\cos 30}}=\sqrt{\frac{1+\cos (2\times 15)}{1-\cos (2\times 15)}}$

$\Rightarrow =\sqrt{\frac{2{\cos }^{2}15}{2{\sin }^{2}15}}$

$\Rightarrow =\sqrt{{\left(\frac{\cos 15}{\sin 15}\right)}^2}$

$\Rightarrow =\cot 15$

OR

$\sqrt{\frac{1+\cos 30}{1-\cos 30}}=\sqrt{\frac{1+\frac{\sqrt{3}}{2}}{1-\frac{\sqrt{3}}{2}}}$

$\Rightarrow =\sqrt{\frac{2+\sqrt{3}}{2-\sqrt{3}}}$