Question

Prove that $\frac{1+\cos A}{\sin A}+\frac{\sin A}{1+\cos A}=2\csc A$

Answer 1

LHS = $\frac{1+cosA}{sinA}+\frac{sinA}{1+cosA}$

$=\frac{(1+cosA{)}^2+si{n}^{2}A}{sinA(1+cosA)}$

$=\frac{1+2cosA+co{s}^{2}A+si{n}^{2}A}{sinA(1+cosA)}$

$=\frac{1+2cosA+1}{sinA(1+cosA)}[si{n}^{2}A+co{s}^{2}A=1]$

$=\frac{2+2cosA}{sinA(1+cosA)}$

$=\frac{2[1+cosA]}{sinA(1+cosA)}$

$=\frac{2}{sinA}$

$=2cosecA=RHS$

LHS = RHS

Hence proved.