Question

Solve :$\frac{\sin {30}^{\circ }+\tan {45}^{\circ }-cosec{60}^{\circ }}{\sec {30}^{\circ }+\cos {60}^{\circ }+\cot {45}^{\circ }}$

Answer 1

$\frac{\sin 30°+\tan 45°-\text{cosec}60°}{\sec 30°+\cos 60°+\cot 45°}$

$=\frac{\frac{1}{2}+1-\frac{2}{\sqrt{3}}}{\frac{2}{\sqrt{3}}+\frac{1}{2}+1}$

$=\frac{\frac{3}{2}-\frac{2}{\sqrt{3}}}{\frac{3}{2}+\frac{2}{\sqrt{3}}}$

$=\frac{\frac{3}{2}-\frac{2}{\sqrt{3}}}{\frac{3}{2}+\frac{2}{\sqrt{3}}}\times \frac{\frac{3}{2}-\frac{2}{\sqrt{3}}}{\frac{3}{2}-\frac{2}{\sqrt{3}}}$ [Rationalising the denominator]

$=\frac{{(\frac{3}{2}-\frac{2}{\sqrt{3}})}^2}{\frac{9}{4}-\frac{4}{3}}$

$=\frac{\frac{9}{4}+\frac{4}{3}-2\times \frac{3}{2}\times \frac{2}{\sqrt{3}}}{\frac{11}{12}}$

$=\frac{\frac{43}{12}-2\sqrt{3}}{\frac{11}{12}}$

$=\frac{43-24\sqrt{3}}{11}$