Simplify \(\frac{\cos 45^{\circ }}{\sec 30^{\circ}+ cosec30^{\circ}}\)

Question

Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 01 Dec 2022 Shift 1)

Answer 1

Given:

\(\frac{\cos 45^{\circ }}{\sec 30^{\circ}+ cosec30^{\circ}}\)

Concept used:

Trigo

Calculation:

\(\frac{\cos 45^{\circ }}{\sec 30^{\circ}+ cosec30^{\circ}}\)

⇒ \(\frac {\frac {1}{\sqrt2}} {\frac {2}{\sqrt3}+ \frac {2}{1}}\)

⇒ \(\frac {\frac {1}{\sqrt2}} {2(\frac {\sqrt3 + 1}{\sqrt3})}\)

⇒ \(\frac {\sqrt3} {2{\sqrt2}({\sqrt3 + 1})}\)

⇒ \(\frac {\sqrt3({\sqrt3 - 1})} {2{\sqrt2}({\sqrt3 + 1})({\sqrt3 - 1})}\)

⇒ \(\frac {\sqrt3({\sqrt3 - 1})} {2{\sqrt2}({3 - 1)}}\)

⇒ \(\frac {({3 - \sqrt3})} {4{\sqrt2}}\)

⇒ \(\frac {({3\sqrt2 - \sqrt6})} {8}\)

∴ The required answer is \(\frac {({3\sqrt2 - \sqrt6})} {8}\).