Question

find the value of 2 sec ^2 theta - 3 cosec ^2 theta ,if cos theta + sin theta / cos theta - sin theta = 1+√3 / 1-√3​

Answer 1

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

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

$\large\underline{\sf{To\:Find - }}$

$\rm :\longmapsto\:2 {sec}^{2}\theta \: - 3 {cosec}^{2}\theta \:$

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

$\boxed{ \sf \: tanx = \dfrac{sinx}{cosx}}$

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

$\boxed{ \sf \: cosec60\degree = \dfrac{2}{ \sqrt{3}}}$

$\boxed{ \sf \: sec60\degree = 2}$

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

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

$\rm :\longmapsto\:\dfrac{cos\theta \: \bigg(1 \: + \: \dfrac{sin\theta \:}{cos\theta \:} \bigg) }{cos\theta \: \bigg(1 \: - \: \dfrac{sin\theta \:}{cos\theta \:} \bigg)} = \dfrac{ \sqrt{3} + 1}{ \sqrt{3} - 1}$

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

On comparing, we get

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

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

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

Now,

Consider,

$\rm :\longmapsto\:2 {sec}^{2}\theta \: - 3 {cosec}^{2}\theta \:$

$\sf \: \: \: = \: \: \: \:2 {sec}^{2}60\degree \: - 3 {cosec}^{2}60\degree\:$

$\sf \: \: \: = \: \: \: 2 \times {(2)}^{2} - 3 \times {\bigg(\dfrac{2}{ \sqrt{3} } \bigg) }^{2}$

$\sf \: \: \: = \: \: \: 8 - 3 \times \dfrac{4}{3}$

$\sf \: \: \: = \: \: \: 8 - 4$

$\sf \: \: \: = \: \: \: 4$

Additional Information :-

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

Answer 2

Step-by-step explanation:

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