Question

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

Answer 1

$\frac{{\sin 30}^0+{\tan 45}^0-cosec{60}^0}{{\sec 30}^0+{\cos 60}^0+{\cot 45}^0}$

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

$=\frac{\frac{9}{4}-2.\frac{3}{2}.\frac{2}{\sqrt{3}}+{\left(\frac{2}{\sqrt{3}}\right)}^2}{{\left(\frac{3}{2}\right)}^2-{\left(\frac{2}{\sqrt{3}}\right)}^2}$

$=\frac{\frac{9}{4}-2\sqrt{3}+\frac{4}{3}}{\frac{9}{4}-\frac{4}{3}}=\frac{\frac{27-24\sqrt{3}+16}{12}}{\frac{27-16}{12}}$

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