Question

$\frac{cosecA}{cosecA-1}+\frac{cosecA}{cosecA+1}=2se{c}^{2}A$

Answer 1

Taking LHS;
⇒$\frac{cosecA}{cosecA-1}$ + $\frac{cosecA}{cosecA+1}$

Taking LCM of the denominator;

⇒$\frac{cosecA(cosecA+1)+cosecA(cosecA-1)}{(cosecA+1)(cosecA-1)}$

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

⇒$\frac{2cose{c}^{2}A}{(cose{c}^{2}A-1)}$

⇒$\frac{2cose{c}^{2}A}{(cose{c}^{2}A-1)}$

⇒$\frac{\frac{2}{si{n}^{2}A}}{\frac{1}{si{n}^{2}A}-1}$

⇒$\frac{\frac{2}{si{n}^{2}A}}{\frac{1-si{n}^{2}A}{si{n}^{2}A}}$

⇒$\frac{2}{1-si{n}^{2}A}$

⇒$\frac{2}{co{s}^{2}A}$

⇒$2se{c}^{2}A$

LHS = RHS, hence it is proved.