Question

Prove that $\mathrm{sinA}(1+\mathrm{tanA})+\mathrm{cosA}(1+\mathrm{cotA})=\mathrm{secA}+\mathrm{cosecA}$

Answer 1

As we have $\mathrm{sinA}(1+\mathrm{tanA})+\mathrm{cosA}(1+\mathrm{cotA})=\mathrm{secA}+\mathrm{cosecA}$

Taking LHS $\mathrm{sinA}(1+\mathrm{tanA})+\mathrm{cosA}(1+\mathrm{cotA})$

$$\begin{gathered} \Rightarrow \sin A\left(1+\frac{\mathrm{si}nA}{\cos A}\right)+\cos A\left(1+\frac{\cos A}{\sin A}\right) \\ \Rightarrow \sin A\left(\frac{\cos A+\sin A}{\cos A}\right)+\cos A\left(\frac{\sin A+\cos A}{\sin A}\right) \\ \Rightarrow \left(\sin A+\cos A\right)\left(\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}\right) \\ \Rightarrow \left(\sin A+\cos A\right)\left(\frac{{\sin }^{2}A+{\cos }^{2}A}{\cos A\sin A}\right) \\ \Rightarrow \frac{\left(\sin A+\cos A\right)\left(1\right)}{\cos A\sin A}\left[\because {\sin }^{2}A+{\cos }^{2}A=1\right] \\ \Rightarrow \frac{\sin A}{\cos A\sin A}+\frac{\cos A}{\cos A\sin A} \\ \Rightarrow \frac{1}{\cos A}+\frac{1}{\sin A} \\ \Rightarrow secA+\cos ecA \end{gathered}$$

Hence LHS=RHS proved.