Prove that :

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

To prove:

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

Solving L.H.S. of the above equation :

$\Rightarrow \text{sin A(1 + tan A) + cos A(1 + cot A)} \\[1em] \Rightarrow \text{sin A + sin A tan A + cos A + cos A cot A} \\[1em] \Rightarrow \text{sin A + sin A} \times \dfrac{\text{sin A}}{\text{cos A}} + \text{cos A + cos A} \times \dfrac{\text{cos A}}{\text{sin A}} \\[1em] \Rightarrow \text{sin A + cos A} + \dfrac{\text{sin}^2 A}{\text{cos A}} + \dfrac{\text{cos}^2 A}{\text{sin A}} \\[1em] \Rightarrow \Big(\text{sin A} + \dfrac{\text{cos}^2 A}{\text{sin A}}\Big) + \Big(\text{cos A} + \dfrac{\text{sin}^2 A}{\text{cos A}}\Big) \\[1em] \Rightarrow \Big(\dfrac{\text{sin}^2 A + \text{cos}^2 A}{\text{sin A}}\Big) + \Big(\dfrac{\text{cos}^2 A + \text{sin}^2 A}{\text{cos A}}\Big)$

By formula,

sin² A + cos² A = 1.

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

∴ L.H.S. = R.H.S.

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