Question

if cosec theta =2, find the value of 1/tan theta +sin theta/1+cos theta

Answer 1

cosec∅ = 2
so, sin∅ = 1/2 , cos∅ = √3/2 { when we assume ∅ is an acute angle }
tan∅ = 1/√3

now ,
1/tan∅ + sin∅/(1+cos∅)
= 1/(1/√3) + (1/2)/(1+√3/2)
= √3 + 1/(2+√3)
= (4+2√3)/(2+√3)
= 2(2+√3)/(2+√3)
= 2

Answer 2

Answer = cosec θ = 2.
This is true for all values of θ.

$\frac{cos \theta}{ \sin( \theta) } + \frac{ \sin( \theta) }{ 1 + cos\theta} = \frac{ \cos( \theta) + {cos}^{2} \theta + {sin}^{2} \theta }{sin \theta \: (1 + cos \theta) } \\ \\ = \frac{1 + \cos( \theta) }{(1 + cos \theta) \times \sin( \theta) } \\ \\ = \frac{1}{ \sin( \theta) } = \csc( \theta) = 2$