Question

The value of $\cot 70^{\circ }+4\cos 70^{\circ }$ is

• A. $\frac{1}{\sqrt{3}}$
• B. $\sqrt{3}$
• C. $2\sqrt{3}$
• D. $\frac{1}{2}$

Answer 1

Answer: (B)

$\cot 70^{\circ }+4\cos 70^{\circ }=\frac{\cos 70^{\circ }+4\sin 70^{\circ }\cos 70^{\circ }}{\sin 70^{\circ }}$
$=\frac{\cos 70^{\circ }+2\sin 140^{\circ }}{\sin 70^{\circ }}$
$=\frac{\cos 70^{\circ }+2\sin \left(180^{\circ }-40^{\circ }\right)}{\sin 70^{\circ }}$
$=\frac{\sin 20^{\circ }+\sin 40^{\circ }+\sin 40^{\circ }}{\sin 70^{\circ }}$
$=\frac{2\sin 30^{\circ }\cos 10^{\circ }+\sin 40^{\circ }}{\sin 70^{\circ }}$
$=\frac{\sin 80^{\circ }+\sin 40^{\circ }}{\sin 70^{\circ }}$
$=\frac{2\sin 60^{\circ }\cos 20^{\circ }}{\sin 70^{\circ }}$
$=\sqrt{3}$