\(\sqrt {3}\)cosec20° - sec20° =?

4

\(\frac{\sqrt {3}}{sin20°}\)-\(\frac{1}{cos20°}\)

⇒ \(\frac{cos20°\sqrt {3}-sin20°}{sin20°cos20°}\)

Divide and multiply numerator by 2

⇒ \(\frac{2\frac{\sqrt {3}}{2}cos20°-\frac{1}{2}sin20°}{sin20°cos20°}\)

⇒ \(\frac{2sin60°cos20°- cos60°sin20°}{sin20°cos20°}\)

⇒ Again multiply and divide the eq. by 2

2.\(\frac{2sin60°cos20°- cos60°sin20°}{2sin20°cos20°}\)   [sin(A-B) = sinAcosB - cosAsinB, 2sinAcosB = sin2A]

= 2.\(\frac{2sin(60°-20°)}{sin40°}\)

= 4