SSC CGL 2022 PYQ — Question 31

The correct answer is option 4 i.e. $\frac{7\sqrt{3}}{54}$

⇒ $\frac{sin^2 \theta}{cos^2 \theta-3 cos\theta+2}=1$

⇒ sin²$\theta$ = cos²$\theta$ - 3 cos$\theta$ + 2.

⇒ sin²$\theta$ = cos²$\theta$ - 3 cos$\theta$ + 2( sin²$\theta$ +  cos²$\theta$ )

⇒ sin²$\theta$ + 3cos²$\theta$ - 3cos$\theta$ = 0

⇒ ( 1 + cos$\theta$)(1 - cos$\theta$) + 3 cos$\theta$(cos$\theta$ - 1) = 0

⇒ (1 -  cos$\theta$ )(1 - 2 cos$\theta$) = 0

⇒ cos$\theta$ = 1  and  cos$\theta$ = 1/2

⇒ $\theta$ = 0° and 60°.

⇒ $\frac{tan^2 \frac{\theta}{2}+sin^2\frac{\theta}{2}}{tan \theta+sin\theta}$ = $\frac{tan^2 \frac{60}{2}+sin^2\frac{60}{2}}{tan60 + sin 60}$

⇒ $\frac{(\frac{1}{\sqrt3})^2 + (\frac{1}{2})^2}{\sqrt3 + \frac{\sqrt3}{2}}$ = $\frac{7\sqrt{3}}{54}$.