if tan^(−1) (x)=(1/2)cos^(−1) ((5/(13))) find x


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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
	$let θ = cos^(−1) ((5/(13))) ⇒ cos θ = (5/(13)) ⇒tan^(−1) (x)=(θ/2) ⇒x = tan ((θ/2)) remember tan θ= ((2tan ((θ/2)))/(1−tan^2 ((θ/2)))) ((12)/5) = ((2x)/(1−x^2 )) ⇒6−6x^2 = 5x 6x^2 +5x−6 = 0 (3x−2)(2x+3) = 0 ⇒ { ((x=(2/3))),((x=−(3/2))) :}$
	$let θ = \cos^{- 1} (\frac{5}{13}) \Rightarrow \cos θ = \frac{5}{13}$ $\Rightarrow \tan^{- 1} (x) = \frac{θ}{2} \Rightarrow x = \tan (\frac{θ}{2})$ $remember \tan θ = \frac{2 \tan (\frac{θ}{2})}{1 - \tan^{2} (\frac{θ}{2})}$ $\frac{12}{5} = \frac{2 x}{1 - x^{2}} \Rightarrow 6 - 6 x^{2} = 5 x$ $6 x^{2} + 5 x - 6 = 0$ $(3 x - 2) (2 x + 3) = 0 \Rightarrow {\begin{cases} x = \frac{2}{3} \\ x = - \frac{3}{2} \end{cases}$
	Commented by i jagooll last updated on 18/May/20

	$thank you$
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