Question

prove that;
tan A / 1+sec A - tan A /1- sec A =2 cosec A

Answer 1

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