Question

prove : $\frac{cotA+cosecA-1}{cotA-cosecA+1}=\frac{1+cosA}{sinA}$

Answer 1

$\frac{\cot A+cosecA-1}{\cot A-cosecA+1}=\frac{\frac{\cos A}{\sin A}+\frac{1}{\sin A}-1}{\frac{\cos A}{\sin A}-\frac{1}{\sin A}+1}$

$=\frac{\cos A-\sin A+1}{\cos A+\sin A-1}$

As $\cos A=2{\cos }^{2}A/2-1$

$=1-2{\sin }^{2}A/2$

& $\sin A=2\sin A/2\cos A/2$

$=\frac{\left(2{\cos }^{2}A/2\right)-\left(2\sin A/2\cos A/2\right)}{\left(2\sin A/2\cos A/2\right)-\left(2{\sin }^{2}A/2\right)}$

$=\frac{2\cos A/2(\cos A/2-\sin A/2)}{2\sin A/2(\cos A/2-\sin A/2)}$

$=\cot A/2$

R.H.S $\frac{1+\cos A}{\sin A}=\frac{2{\cos }^{2}A/2}{2\sin A/2\cos A/2}=\cot A/2$