Question

prove that (sinA/(1-cosA))+((1-cosA)/sinA)=2cosec A

Answer 1

$\frac{sinA}{1-cosA} + \frac{1-cosA}{sinA} = 2cosecA$

In LHS,

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

Since $(a-b)^2 = a^2+b^2-2ab$

⇒$\frac{sin^2A+1+cos^2A-2cosA}{1-cosA(sinA)}$

As $sin^2A+cos^2A=1$

⇒$\frac{1+1-2cosA}{1-cosA(sinA)}$

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

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

=$\frac{2}{sinA}$

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

=$2cosecA$

∴ LHS = RHS

Hope I did it right.