Question

solve cos thetha / 1- sin thetha + cos thetha / 1+ sin thetha​ = 4

Answer 1

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

Given trigonometric equation is

$\sf \: \dfrac{cos\theta }{1 - sin\theta } + \dfrac{cos\theta }{1 + sin\theta } = 4$

can be rewritten as

$\sf \:cos\theta \bigg(\dfrac{1}{1 - sin\theta } + \dfrac{1}{1 + sin\theta }\bigg) = 4$

$\sf \:cos\theta \bigg(\dfrac{1 + sin\theta + 1 - sin\theta }{(1 - sin\theta)(1 + sin\theta )} \bigg) = 4$

$\sf \:cos\theta \bigg(\dfrac{2}{1 - {sin}^{2} \theta } \bigg) = 4$

$\sf \: \dfrac{2cos\theta }{ {cos}^{2}\theta } = 4$

$\sf \: \dfrac{2}{cos\theta } = 4$

$\sf \: cos\theta = \dfrac{1}{2}$

$\sf \: cos\theta = cos60 \degree$

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

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Formulae Used :-

$\boxed{ \rm{ \:(x + y)(x - y) = {x}^{2} - {y}^{2} \: }}$

$\boxed{ \rm{ \: {sin}^{2}x + {cos}^{2}x = 1 \: }}$

$\rule{190pt}{2pt}$

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

refer the given attachment