Question

1-costheta/1+costheta = (cosec theta - cottheta)²

Answer 1

$$\begin{gathered} \mathrm{LHS}=\frac{1-\mathrm{cos\theta }}{1+\mathrm{cos\theta }} \\ =\left(\frac{1-\mathrm{cos\theta }}{1+\mathrm{cos\theta }}\right)\times \frac{\left(1-\mathrm{cos\theta }\right)}{\left(1-\mathrm{cos\theta }\right)} \\ =\frac{{\left(1-\mathrm{cos\theta }\right)}^2}{1-{\cos }^2\mathrm{\theta }} \\ =\frac{1+{\cos }^2\mathrm{\theta }-2\mathrm{cos\theta }}{{\sin }^2\mathrm{\theta }} \\ =\frac{1}{{\sin }^2\mathrm{\theta }}+\frac{{\cos }^2\mathrm{\theta }}{{\sin }^2\mathrm{\theta }}-\frac{2\mathrm{cos\theta }}{{\sin }^2\mathrm{\theta }} \\ ={\mathrm{cosec}}^2\mathrm{\theta }+{\cot }^2\mathrm{\theta }-2\mathrm{cosec\theta }.\mathrm{cot\theta } \\ ={\left(\mathrm{cosec\theta }-\mathrm{cot\theta }\right)}^2 \\ =\mathrm{RHS} \end{gathered}$$