NIMCET 2020 — Mathematics PYQ

Concept:
                                               Trigonometric Ratios:

$\tan \theta$=$\frac{\sin \theta }{\cos \theta }$		$\cot \theta =\frac{\cos \theta }{\sin \theta }$
$\csc \theta =\frac{1}{\sin \theta }$	$\sec \theta =\frac{1}{\cos \theta }$		$\cot \theta =\frac{1}{\tan \theta }$
${\sin }^2\theta +{\cos }^2\theta =1$

Calculation :

Let $$\tan A=m\Rightarrow \cot A=\frac{1}{m}$$  

$$\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}$$  

$$=\frac{m}{1-\frac{1}{m}}+\frac{\frac{1}{m}}{1-m}$$  

$$=\frac{m^2}{m-1}-\frac{1}{m(m-1)}$$  

$$=\frac{m^3-1}{m(m-1)}$$  

$$=\frac{(m-1)(m^2+m+1)}{m(m-1)}$$  

$$=m^2+m+1$$  

$$=m+1+\frac{1}{m}$$  

$$=\left(\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}\right)+1$$  

$$=\left(\frac{{\sin }^{2}A+{\cos }^{2}A}{\sin A\cos A}\right)+1$$  

$$=\frac{1}{\sin A\cos A}+1$$  

$$=\sec A\csc A+1$$