Question

(1 + CotA - Cosec) (1+tan A+sec A)= 2 ​

Answer 1

Solution:

Given:-

$\rm\implies ( 1+\cot A-\csc A)(1+\tan A +\sec A)$

Using trigonometry identities

$\rm \implies \cot A =\dfrac{\cos A}{\sin A}$

$\rm \implies \tan A=\dfrac{\sin A}{\cos A}$

$\rm \implies \sec A=\dfrac{1}{\cos A}$

$\rm \implies \csc A=\dfrac{1}{\sin A}$

By applying this identities we get

$\rm\implies\bigg(1+\dfrac{\cos A}{\sin A} -\dfrac{1}{\sin A} \bigg)\bigg(1+\dfrac{\sin A}{\cos A} +\dfrac{1}{\cos A} \bigg)$

Taking lcm we get

$\rm\implies\bigg(\dfrac{\sin A+\cos A-1}{\sin A} \bigg)\bigg(\dfrac{\cos A+\sin A +1}{\cos A} \bigg)$

Now multiply

$\rm\implies\dfrac{\sin A\cos A+\sin^{2}A+\sin A+\cos^{2} A+\sin A\cos A+\cos A-\cos A-\sin A-1 }{\sin A.\cos A}$

$\rm\implies\dfrac{\sin A\cos A+1+\sin A\cos A-1 }{\sin A.\cos A}$

$\rm\implies\dfrac{2\sin A\cos A}{\sin A.\cos A}$

$\implies 2$

Hence proved