Question

$\frac{\cot A+\text{cosec}A-1}{\cot A-\text{cosec}A+1}$ is

• A. $\frac{1-\cos A}{\sin A}$
• B. $\frac{1+\sin A}{\cos A}$
• C. $\frac{1-\sin A}{\cos A}$
• D. $\frac{1+\cos A}{\sin A}$

Answer 1

The correct option is D $\frac{1+\cos A}{\sin A}$

$\frac{\cot A+cosecA-1}{\cot A-cosecA+1}$

$\Rightarrow \frac{\cot A+cosecA-(cose{c}^{2}A-{\cot }^{2}A)}{\cot A-cosecA+1}$

$\Rightarrow \frac{(\cot A+cosecA)-(cosecA-\cot A)(cosecA+\cot A)}{\cot A-cosecA+1}$

$\Rightarrow \frac{(cosecA+\cot A)(1-cosecA+\cot A)}{\cot A-cosecA+1}$

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

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

Hence, the answer is $\frac{1+\cos A}{\sin A}.$