Question

Prove that $\frac{\cos A+\sin A-1}{\cos A-\sin A+1}=\frac{1}{\text{cosec}A+\cot A}$, using the identity ${\text{cosec}}^{2}A-{\cot }^{2}A=1.$

Answer 1

We have, $\frac{cosA+sinA-1}{cosA-(sinA-1)}\times \frac{cosA+(sinA-1)}{cosA+(sinA+1)}$

$\Rightarrow \frac{co{s}^{2}A+(sinA-1{)}^2+2cosA(sin-1)}{co{s}^{2}A-(sinA-1{)}^2}$

$\Rightarrow \frac{co{s}^{2}A+si{n}^{2}A+12sinA+2cosAsinA-2cosA}{co{s}^{2}A-si{n}^{2}A-1+2sinA}$

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

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

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

$\Rightarrow \frac{1}{cosecA+cotA}$