Question

Prove the following identity:
xi $\frac{\cot A+cosecA–1}{\cot A–cosecA+1}=\frac{\cos A+1}{\sin A}$

Answer 1

Proving LHS = RHS.

From the question first we consider Left Hand Side (LHS),

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

We know that, $cose{c}^{2}A–{\cot }^{2}A=1$.

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

Also we know that,$(a^2–b^2)=(a+b)(a–b)$.

$=\frac{(\cot A+cosecA)–(cosecA–\cot A)(cosecA+\cot A)}{\cot A–cosecA+1}$

$=\frac{(\cot A+cosecA)(1–cosecA+\cot A)}{(\cot A–cosecA+1)}$

$=\cot A+cosecA$

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

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

Then, Right Hand Side $=\frac{\cos A+1}{\sin A}$ .

Therefore,LHS = RHS.

Hence proved.