Question

Evaluate 4 cos ${20}^{\circ }$ - $\sqrt{3}cot{20}^{\circ }$ and

Answer 1

$L.H.S=\frac{1}{sin{20}^{\circ }}(4cos{20}^{\circ }sin{20}^{\circ }-\sqrt{3}cos{20}^{\circ })$

$L.H.S=\frac{2}{sin{20}^{\circ }}(sin{40}^{\circ }-\frac{\sqrt{3}}{2}cos{20}^{\circ })$

$L.H.S=\frac{1}{sin{20}^{\circ }}(2sin{40}^{\circ }-2sin{60}^{\circ }cos{20}^{\circ })$

$L.H.S=\frac{1}{sin{20}^{\circ }}\left[\right(2sin{40}^{\circ }-(sin{80}^{\circ }+sin{40}^{\circ })]$

$L.H.S=\frac{1}{sin{20}^{\circ }}[sin{40}^{\circ }-sin{80}^{\circ }]$

$L.H.S=\frac{1}{sin{20}^{\circ }}\left[2sin\right(-{20}^{\circ }\left)cos{60}^{\circ }\right]=-1$