Prove the following
Question:

If $\sin A=\frac{1}{2}$ then the value of $\cot A$ is

(a) $\sqrt{3}$

(b) $\frac{1}{\sqrt{3}}$

(c) $\frac{\sqrt{3}}{1}$

(d) 1

Solution:

(a) Given, $\sin A=\frac{1}{2}$

$\therefore \quad \cos A=\sqrt{1-\sin ^{2} A}=\sqrt{1-\left(\frac{1}{2}\right)^{2}}$

$=\sqrt{1-\frac{1}{4}}=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}$ $\left[\because \sin ^{2} A+\cos ^{2}=1 \Rightarrow \cos A=\sqrt{1-\sin ^{2} A}\right]$

Now, $\cot A=\frac{\cos A}{\sin A}=\frac{\frac{\sqrt{3}}{2}}{1}=\sqrt{3}$