Question

$\frac{1+{\tan }^2{30}^o}{1-{\tan }^2{30}^o}+{\csc }^2{60}^0-{\cos }^2{45}^o+{\sin }^2{45}^o+\frac{1+{\cot }^2{60}^o}{1-{\cot }^2{60}^o}$

Answer 1

We have,

$\frac{1+{\tan }^2{30}^{\circ }}{1-{\tan }^2{30}^{\circ }}+{\csc }^2{60}^{\circ }-{\cos }^2{45}^{\circ }+{\sin }^2{45}^{\circ }+\frac{1+{\cot }^2{60}^{\circ }}{1-{\cot }^2{60}^{\circ }}$

$\Rightarrow \frac{1+{\left(\frac{1}{\sqrt{3}}\right)}^2}{1-{\left(\frac{1}{\sqrt{3}}\right)}^2}+{\left(\frac{2}{\sqrt{3}}\right)}^2-{\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{\sqrt{2}}\right)}^2+\frac{1+{\left(\frac{1}{\sqrt{3}}\right)}^2}{1-{\left(\frac{1}{\sqrt{3}}\right)}^2}$

$\Rightarrow \frac{1+\left(\frac{1}{3}\right)}{1-\left(\frac{1}{3}\right)}+\left(\frac{4}{3}\right)+\frac{1+\left(\frac{1}{3}\right)}{1-\left(\frac{1}{3}\right)}$

$\Rightarrow \frac{4}{2}+\frac{4}{3}+\frac{4}{2}$

$\Rightarrow 2+\frac{4}{3}+2$

$\Rightarrow 4+\frac{4}{3}$

$\Rightarrow \frac{16}{3}$

Hence, this is the answer.