Question

Prove that
SinA/1+CosA + 1+CosA/SinA = 2CosecA

Answer 1

$\textit{\underline{\underline{Proof...}}}$

$\displaystyle\ \textsf{LHS...}\\ \\ \\ \Rightarrow\ \frac{\sin A}{1+\cos A}+\frac{1+\cos A}{\sin A} \\ \\ \\ \Rightarrow\ \frac{\sin^2A+(1+\cos A)^2}{(1+\cos A)\sin A} \\ \\ \\ \Rightarrow\ \frac{\sin^2A+1+2\cos A+\cos^2A}{\sin A(\cos A+1)} \\ \\ \\ \Rightarrow\ \frac{\sin^2A+\cos^2A+2\cos A+1}{\sin A(\cos A+1)} \\ \\ \\ \Rightarrow\ \frac{1+2\cos A+1}{\sin A(\cos A+1)}$

$\displaystyle \Rightarrow\ \frac{2\cos A+2}{\sin A(\cos A+1)} \\ \\ \\ \Rightarrow\ \frac{2(\cos A+1)}{\sin(\cos A+1)} \\ \\ \\ \Rightarrow\ \frac{2}{\sin A} \\ \\ \\ \Rightarrow\ 2 \cdot \frac{1}{\sin A} \\ \\ \\ \Rightarrow\ 2\csc A \\ \\ \\ \Rightarrow\ \textsf{...RHS}$

$\large \textsc{Hence Proved!!!}$