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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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