prove that tan-11/4 +tan-1 2/9 =1/2tan-1 4/3 - Maths - Inverse Trigonometric Functions

Question

prove that tan-11/4 +tan-1 2/9
=1/2tan-1 4/3

Ajanta Trivedi · Accepted Answer

tan-11/4 +tan-1 2/9 =1/2tan-1 4/3 to prove the above we will prove that $2 \tan^{- 1} \frac{1}{4} + 2 \tan^{- 1} \frac{2}{9} = \tan^{- 1} \frac{4}{3}$ let $A = 2 \tan^{- 1} \frac{1}{4}, B = 2 \tan^{- 1} \frac{2}{9}, \Rightarrow \tan \frac{A}{2} = \frac{1}{4}, \tan \frac{B}{2} = \frac{2}{9}$ then $\tan (\frac{A}{2} + \frac{B}{2}) = \frac{\tan \frac{A}{2} + \tan \frac{B}{2}}{1 - \tan \frac{A}{2} \cdot \tan \frac{B}{2}} = \frac{\frac{1}{4} + \frac{2}{9}}{1 - \frac{1}{4} \cdot \frac{2}{9}} = \frac{17}{36} \cdot \frac{36}{34} = \frac{1}{2}$ therefore $\frac{A + B}{2} = \tan^{- 1} \frac{1}{2} \Rightarrow A + B = 2 \tan^{- 1} \frac{1}{2}$ (1) let A+B=C let $C = 2 \tan^{- 1} \frac{1}{2} \Rightarrow \frac{C}{2} = \tan^{- 1} \frac{1}{2} \Rightarrow \tan \frac{C}{2} = \frac{1}{2}$ $\tan C = \frac{2 \tan \frac{C}{2}}{1 - \tan^{2} \frac{C}{2}} = \frac{2 * \frac{1}{2}}{1 - \frac{1}{4}} = \frac{1}{3 / 4} = \frac{4}{3}$ $\Rightarrow C = \tan^{- 1} \frac{4}{3} (2)$ hence A+B=C i e $2 \tan^{- 1} \frac{1}{4} + 2 \tan^{- 1} \frac{2}{9} = \tan^{- 1} \frac{4}{3}$ $\Rightarrow$ $\tan^{- 1} \frac{1}{4} + \tan^{- 1} \frac{2}{9} = \frac{1}{2} \cdot \tan^{- 1} \frac{4}{3}$ which is the required result