Question

Find the value of $\frac{\cos A-\sin A+1}{\cos A+\sin A-1}-(\text{cosec}A+\cot A)$

Answer 1

Now,

$\frac{\cos A-\sin A+1}{\cos A+\sin A-1}-(\text{cosec}A+\cot A)$

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

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

$=\frac{({\cos }^{2}A+2\cos A+1)-\left({\sin }^{2}A\right)}{2\sin A.\cos A}-(\text{cosec}A+\cot A)$ [ Since ${\sin }^{2}A+{\cos }^{2}A=1$]

$=\frac{1+\cos 2A+2\cos A}{2\sin A.\cos A}-(\text{cosec}A+\cot A)$ [ Since ${\cos }^{2}A-{\sin }^{2}A=\cos 2A$]

$=\frac{2{\cos }^{2}A+2\cos A}{2\sin A.\cos A}-(\text{cosec}A+\cot A)$ [ Since $1+\cos 2A=2{\cos }^{2}A$]

$=\cot A+\text{cosec}A-(\cot A+\text{cosec}A)$

$=0$.