SSC CGL 2024 PYQ — Question 53

The correct answer is option 4 i.e.  tan   $\frac{θ}{2}$

${{1 + sinθ - cosθ} \over 1 + sinθ + cosθ}$

$=\big[{{1 + sinθ - cosθ} \over {1 + sinθ + cosθ}}\big] \times \big[{{1 + sinθ - cosθ} \over {1 + sinθ - cosθ}}\big]$

$= {{(1 + sinθ - cosθ)^2} \over {(1 + sinθ)^2 - cos^2θ}}$

= (1 + sin²θ + cos²θ + 2sinθ - 2cosθ - 2sinθcosθ)/(1 + 2sinθ + sin²θ - cos²θ)

= (2 + 2sinθ - 2cosθ - 2sinθcosθ)/[1 + 2sinθ + sin²θ - (1 - sin²θ)]

= [2(1 + sinθ - cosθ - sinθcosθ)]/(1 + 2sinθ + sin²θ - 1 + sin²θ)

= 2[(1 + sinθ) - cosθ(1 + sinθ)]/(2sinθ + 2sin²θ)

= 2[(1 + sinθ) - cosθ(1 + sinθ)]/[2sinθ(1 + sinθ)]

= [2(1 + sinθ)(1 - cosθ)]/[2sinθ(1 + sinθ)]

= (1 - cosθ)/sinθ

= [2sin²(θ/2)]/[2sin(θ/2)cos(θ/2)]

= sin(θ/2)/cos(θ/2)

= tan(θ/2)

Therefore, the value of the given expression is tan(θ/2).