Question

$\frac{cosA-sinA+1}{cosA+sin-1}=cosecA+cotA,$, using the identity $cose{c}^{2}A=1+co{t}^{2}A$.

Answer 1

We have,

L.H.S.

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

On rationalize and we get,

$\frac{\cos A-\sin A+1}{(\cos A+\sin A)-1}\times \frac{(\cos A+\sin A)+1}{(\cos A+\sin A)+1}$

$=\frac{{\cos }^{2}A+\cos A\sin A+\cos A-\sin A\cos A-{\sin }^{2}A-\sin A+\cos A+\sin A+1}{{(\cos A+\sin A)}^2-1^2}$

$=\frac{{\cos }^{2}A-{\sin }^{2}A+2\cos A+1}{{\cos }^{2}A+{\sin }^{2}A+2\sin A\cos A-1}$

$=\frac{{\cos }^{2}A+2\cos A+1-{\sin }^{2}A}{1+2\sin A\cos A-1}$

$=\frac{2{\cos }^{2}A+2\cos A}{2\sin A\cos A}$

$=\frac{2\cos A(\cos A+1)}{2\sin A\cos A}$

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

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

$=\mathrm{cscA}+cotA$

R.H.S.