Question

Prove that $\frac{\cot A+\csc A-1}{\cot A-\csc A+1}=\frac{1+\cos A}{\sin A}$

Answer 1

Consider the $LHS$.

$\Rightarrow \frac{\cot A+\csc A-1}{\cot A-\csc A+1}$

We know that,

${\csc }^{2}A-{\cot }^{2}A=1$

$a^2-b^2=(a-b)(a+b)$

Therefore,

$\Rightarrow \frac{\cot A+\csc A-({\csc }^{2}A-{\cot }^2)}{\cot A-\csc A+1}$

$\Rightarrow \frac{\cot A+\csc A[1-\csc A+\cot A]}{\cot A-\csc A+1}$

$\Rightarrow \cot A+\csc A$

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

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

Hence, $LHS=RHS$.