Question

cottheta-1/ cottheta+1 = 1-√3/ 1+√3 find the acute angle theta​

Answer 1

$\begin{gathered}\Large{\bold{{\underline{Formula \: Used - }}}}  \end{gathered}$

$\boxed{ \sf \: cotx = \dfrac{1}{tanx}}$

$\boxed{ \sf \: tan60 \degree \: = \sqrt{3}}$

$\boxed{ \sf \: cot60\degree = \dfrac{1}{ \sqrt{3}}}$

$\large\underline{\sf{Solution-}}$

Method :- 1

$\rm :\longmapsto\:\dfrac{cot \theta \: - 1 }{cot\theta \: + 1} = \dfrac{1 - \sqrt{3} }{1 + \sqrt{3} }$

$\rm :\longmapsto\:\dfrac{\dfrac{1}{tan\theta} - 1 }{\dfrac{1}{tan\theta} + 1} = \dfrac{1 - \sqrt{3} }{1 + \sqrt{3} }$

$\rm :\longmapsto\:\dfrac{\dfrac{1 - tan\theta}{ \cancel{tan\theta}}}{ \: \: \dfrac{1 + tan\theta}{ \cancel{tan\theta}} \: \: } = \dfrac{1 - \sqrt{3} }{ \: \: 1 + \sqrt{3} \: \: }$

$\rm :\longmapsto\:\dfrac{1 - tan\theta}{1 + tan\theta} = \dfrac{1 - \sqrt{3} }{1 + \sqrt{3} }$

So, on comparing we get

$\rm :\longmapsto\:tan\theta \: = \: \sqrt{3}$

$\rm :\longmapsto\:tan\theta \: = \: tan60\degree$

$\bf\implies \:\theta \: = \: 60\degree$

Method:- 2

$\rm :\longmapsto\:\dfrac{cot\theta \: - 1 }{cot\theta \: + 1} = \dfrac{1 - \sqrt{3} }{1 + \sqrt{3} }$

$\rm :\longmapsto\:(cot\theta - 1)(1 + \sqrt{3}) = (cot\theta + 1)(1 - \sqrt{3})$

$\rm :\longmapsto\:cot\theta + \sqrt{3}cot\theta - 1 - \sqrt{3} = cot\theta + 1 - \sqrt{3} - \sqrt{3}cot\theta$

$\rm :\longmapsto\: \sqrt{3}cot\theta - 1 = 1 - \sqrt{3}cot\theta$

$\rm :\longmapsto\:2 \sqrt{3}cot\theta = 2$

$\rm :\longmapsto\:cot\theta = \dfrac{1}{ \sqrt{3} }$

$\rm :\longmapsto\:cot\theta = cot60\degree$

$\bf\implies \:\theta = 60\degree$

Additional Information :-

$\boxed{ \sf \: cosecx = \dfrac{1}{sinx}}$

$\boxed{ \sf \: secx = \dfrac{1}{cosx}}$

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$