Question

Prove that 1+tan A
/ 2 sin A
+

1+cot A
/ 2 cos A
= cosec A + sec A .​

Answer 1

$\frac{1+\tan A}{2\sin A}+\frac{1+\cot A}{2\cos A}=\frac{1}{2}\left(\frac{1+\tan A}{\sin A}+\frac{1+\cot A}{\cos A}\right)\\\\=\frac{\left(1+\tan A\right)\cos A+\left(1+\cot A\right)\sin A}{2\sin A\cos A}\\=\frac{\left(1+\frac{\sin A}{\cos A}\right)\cos A+\left(1+\frac{\cos A}{\sin A}\right)\sin A}{2\sin A\cos A}\\\\=\frac{\left(\cos A+\sin A\right)+\left(\sin A+\cos A\right)}{2\sin A\cos A}\\\\=\frac{2\sin A+2\cos A}{2\sin A\cos A}\\\\=\frac{2\sin A}{2\sin A\cos A}+\frac{2\cos A}{2\sin A\cos A}$

$=\frac{1}{\cos A}+\frac{1}{\sin A} \\= cosec A + \sec A$

Hence, proved