Question
Prove that:
$\frac{1+\tan A}{2 \sin A}+\frac{1+\cot A}{2 \cos A}=\operatorname{cosec} A+\sec A$
$\frac{1+\tan A}{2 \sin A}+\frac{1+\cot A}{2 \cos A}=\operatorname{cosec} A+\sec A$
✓
Answer
LHS $=\frac{1+\tan A}{2 \sin A}+\frac{1+\cot A}{2 \cos A}$
$=\frac{\cos A+\sin A}{2 \sin A \cos A}+\frac{\sin A+\cos A}{2 \cos A \sin A}$
$=\frac{2(\cos A+\sin A)}{2 \sin A \cos A}$
= cosec A + sec A
= RHS
$=\frac{\cos A+\sin A}{2 \sin A \cos A}+\frac{\sin A+\cos A}{2 \cos A \sin A}$
$=\frac{2(\cos A+\sin A)}{2 \sin A \cos A}$
= cosec A + sec A
= RHS
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