Question

Prove tan(θ / 2) = sin θ / (1 + cos θ) for θ in quadrant 1 .

Answer 1

Explanation:

We have to prove the identity
`tan((\Theta )/(2))=(sin\Theta)/(1+cos\Theta )`

We will take right hand side of the identity

`(sin\Theta)/(1+cos\Theta)=(2sin((\Theta )/(2))cos((\Theta )/(2)))/(1+[2cos^(2)((\Theta )/(2))-1])`

`=(2sin((\Theta )/(2))cos((\Theta )/(2)))/(2cos^(2)((\Theta )/(2)))`
`=(sin((\Theta )/(2)))/(cos((\Theta )/(2)))`

`=tan((\Theta )/(2))` [ Tan θ will be positive since θ lies in 1st quadrant ]

= L. H. S.

Hence proved.