Prove that:$ \frac{1-\cos A}{1+\cos A}=(\cot A-\operatorname{cosec} A)^{2} $

To do:

We have to prove that \( \frac{1-\cos A}{1+\cos A}=(\cot A-\operatorname{cosec} A)^{2} \).

Solution:

We know that,

$\sin ^{2} A+\cos^2 A=1$.......(i)

$\operatorname{cosec} A=\frac{1}{\sin A}$......(ii)

$\cot A=\frac{\cos A}{\sin A}$......(iii)

Let us consider RHS,

$(\cot A-\operatorname{cosec} A)^{2}=\left(\frac{\cos A}{\sin A}-\frac{1}{\sin A}\right)^{2}$

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

$=\frac{(\cos A-1)^{2}}{\sin ^{2} A}$

$=\frac{[-(1-\cos A)]^{2}}{1-\cos ^{2} A}$

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

$=\frac{(1-\cos A)(1-\cos A)}{(1+\cos A)(1-\cos A)}$

$=\frac{1-\cos A}{1+\cos A}$

$=$ LHS

Hence proved.      

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Updated on: 10-Oct-2022

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