if-tan-1-x-1-2-cos-1-5-13-find-x-

Question Number 94344 by i jagooll last updated on 18/May/20

if \tan^{- 1} (x) = \frac{1}{2} \cos^{- 1} (\frac{5}{13})

find x

Answered by john santu last updated on 18/May/20

let θ = \cos^{- 1} (\frac{5}{13}) \Rightarrow \cos θ = \frac{5}{13}

\Rightarrow \tan^{- 1} (x) = \frac{θ}{2} \Rightarrow x = \tan (\frac{θ}{2})

remember \tan θ = \frac{2 \tan (\frac{θ}{2})}{1 - \tan^{2} (\frac{θ}{2})}

\frac{12}{5} = \frac{2 x}{1 - x^{2}} \Rightarrow 6 - 6 x^{2} = 5 x

6 x^{2} + 5 x - 6 = 0

(3 x - 2) (2 x + 3) = 0 \Rightarrow {\begin{cases} x = \frac{2}{3} \\ x = - \frac{3}{2} \end{cases}

Commented by i jagooll last updated on 18/May/20

thank you