Prove the following identity:

sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A

Solving L.H.S of equation,

⇒ sin A (1 + tan A) + cos A (1 + cot A)

⇒ sin A + sin A tan A + cos A + cos A cotA

$\Rightarrow \sin A + \dfrac{\sin^2 A}{\cos A} + \cos A + \dfrac{\cos^2 A}{\sin A} \\[1em] \Rightarrow \sin A + \dfrac{\cos^2 A}{\sin A} + \cos A +\dfrac{\sin^2 A}{\cos A} \\[1em] \Rightarrow \dfrac{\sin^2 A + \cos^2 A}{\sin A} + \dfrac{\sin^2 A + \cos^2 A}{\cos A} \\[1em] \Rightarrow \dfrac{1}{\sin A} + \dfrac{1}{\cos A} \\[1em] \Rightarrow \sec A + \cosec A.$

Since, L.H.S. = R.H.S.,

Hence, proved that sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A.