Question

If tan theta = 1/2 then evaluate (cos theta)/(sin theta)+(sin theta)/(1+cos theta) is ?




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Answer 1

Answer:

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Answer 2

GIVEN :–

$\\ \bf \: { \huge{.}} \:\: \tan( \theta) = \dfrac{1}{2}$

TO FIND :–

$\\ \bf \: { \huge{.}} \:\: \dfrac{ \cos( \theta) }{ \sin( \theta) } + \dfrac{ \sin( \theta) }{1 + \cos( \theta) } = ?$

SOLUTION :–

$\\ \bf \implies \tan( \theta) = \dfrac{1}{2}$

• We know that –

$\\ \bf \implies { \sec}^{2}\theta =1 + \tan^{2} ( \theta)$

$\\ \bf \implies { \sec}^{2}\theta =1 + \left( \dfrac{1}{2} \right)^{2}$

$\\ \bf \implies { \sec}^{2}\theta = \left( \dfrac{5}{4} \right)$

$\\ \bf \implies { \sec}\theta = \dfrac{ \sqrt5}{2}$

• Now –

$\\ \bf \:\:=\:\: \dfrac{ \cos( \theta) }{ \sin( \theta) } + \dfrac{ \sin( \theta) }{1 + \cos( \theta) }$

• We should write this as –

$\\ \bf \:\:=\:\: \dfrac{1}{ \dfrac{\sin( \theta)}{ \cos( \theta) }} + \dfrac{ \dfrac{ \sin( \theta)}{ \cos( \theta) } }{ \dfrac{1 + \cos( \theta)}{ \cos( \theta) } }$

$\\ \bf \:\:=\:\: \dfrac{1}{\tan( \theta)} + \dfrac{ \tan( \theta) }{ \sec( \theta) + 1 }$

• Put the values —

$\\ \bf \:\:=\:\: \dfrac{1}{ \dfrac{1}{2} } + \dfrac{ \dfrac{1}{2} }{ \dfrac{ \sqrt{5} }{2} + 1 }$

$\\ \bf \:\:=\:\: 2+ \dfrac{ \dfrac{1}{2} }{ \dfrac{ \sqrt{5} + 2}{2} }$

$\\ \bf \:\:=\:\: 2+ \dfrac{1}{ \sqrt{5} + 2}$

$\\ \bf \:\:=\:\: \dfrac{2( \sqrt{5} + 2) + 1}{ \sqrt{5} + 2}$

$\\ \bf \:\:=\:\: \dfrac{2\sqrt{5} + 4+ 1}{ \sqrt{5} + 2}$

$\\ \bf \:\:=\:\: \dfrac{2\sqrt{5} +5}{ \sqrt{5} + 2}$

$\\ \bf \:\:=\:\: \dfrac{\sqrt{5} (2+\sqrt{5})}{ \sqrt{5} + 2}$

$\\ \bf \:\:=\:\: \sqrt{5}$

$\\ \rule{220}{2}$