Prove. (1 + A - ec A)(1+ A + A) = 2 - Vidyadip

Question

Prove.
$(1 + \cot A - \cos ec A)(1+ \tan A + \sec A) = 2$

✓

Answer

$\text { LHS }=(1+\cot A-\operatorname{cosec} A)(1+\tan A+\sec A)$
$=\left(1+\frac{\cos A}{\sin A}-\frac{1}{\sin A}\right)\left(1+\frac{\sin A}{\cos A}-\frac{1}{\cos A}\right)$
$=\left(\frac{\sin A+\cos A-1}{\sin A}\right)\left(\frac{\cos A+\sin A+1}{\cos A}\right)$
$=\frac{(\sin A+\cos A-1)(\sin A+\cos A+1)}{\sin A \cos A}$
$=\frac{(\sin A+\cos A)^2-(1)^2}{\sin A \cos A}$
$=\frac{\sin ^2 A+\cos ^2 A+2 \sin A \cos A-1}{\sin A \cos A}$
$=\frac{1+2 \sin A \cos A-1}{\sin A \cos A}$
$=\frac{2 \sin A \cos A}{\sin A \cos A}=2=\text { RHS }$

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