Question
Prove that: $\frac{\cot A-\cos A}{\cot A+\cos A}$$=\frac{\ cosec A-1}{\ cosec A+1}$
✓
Answer
LHS = $\frac{\cot A-\cos A}{\cot A+\cos A}$
$=\frac{\frac{\cos A}{\sin A}-\cos A}{\frac{\cos A}{\sin A}+\cos A}$
$=\frac{\frac{\cos A-\sin A \cos A}{\sin A}}{\frac{\cos A+\sin A \cos A}{\sin A}}$
$=\frac{\cos A(1-\sin A)}{\cos A(1+\sin A)}$
$=\frac{1-\sin A}{1+\sin A}$
$=\frac{\frac{1}{\sin A}-1}{\frac{1}{\sin A}+1}$
$=\frac{\ cosec A-1}{\ cosec A+1}$ = RHS
$=\frac{\frac{\cos A}{\sin A}-\cos A}{\frac{\cos A}{\sin A}+\cos A}$
$=\frac{\frac{\cos A-\sin A \cos A}{\sin A}}{\frac{\cos A+\sin A \cos A}{\sin A}}$
$=\frac{\cos A(1-\sin A)}{\cos A(1+\sin A)}$
$=\frac{1-\sin A}{1+\sin A}$
$=\frac{\frac{1}{\sin A}-1}{\frac{1}{\sin A}+1}$
$=\frac{\ cosec A-1}{\ cosec A+1}$ = RHS
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