Question

proove that cot²/(1-cosec)²=1+sin/1-sin

Answer 1

$$\begin{gathered} \mathrm{LHS}=\frac{{\cot }^2\mathrm{A}}{{\left(1-\mathrm{cosecA}\right)}^2} \\ =\frac{{\displaystyle \frac{{\cos }^2\mathrm{A}}{{\sin }^2\mathrm{A}}}}{{\left(1-{\displaystyle \frac{1}{\mathrm{sinA}}}\right)}^2}=\frac{{\cos }^2\mathrm{A}}{{\left(\mathrm{sinA}-1\right)}^2}=\frac{1-{\sin }^2\mathrm{A}}{{\left(1-\mathrm{sinA}\right)}^2} \\ =\frac{\left(1-\mathrm{sinA}\right)\left(1+\mathrm{sinA}\right)}{{\left(1-\mathrm{sinA}\right)}^2}=\frac{\left(1+\mathrm{sinA}\right)}{1-\mathrm{sinA}}=\mathrm{RHS} \end{gathered}$$