Q62 / 160 · TS EAMCET 2022 (19 Jul Shift 2) · Mathematics

If $\theta$ does not lie in the second quadrant and $\theta=\frac{-3}{4}$, then $\tan \frac{\theta}{2}+\sin 2 \theta=$

  • A
    $\frac{97}{75}$
  • Correct answer
    B
    $\frac{-97}{75}$
  • C
    $\frac{-47}{75}$
  • D
    $\frac{47}{75}$

Solution

Given $\tan \theta=\frac{-3}{4}$ $\begin{aligned} & \Rightarrow \sin \theta=\frac{-3}{5}, \cos \theta=\frac{4}{5} \Rightarrow \cos 2 \theta=2 \times \frac{-3}{5} \times \frac{4}{5}=\frac{-24}{25} \\ & \tan \frac{\theta}{2}=\frac{-1}{3} \Rightarrow \tan \frac{\theta}{2}+\sin 2 \theta=\frac{-1}{3}-\frac{24}{25}=\frac{-97}{75}\end{aligned}$

Hence, the correct answer is option: 'B'.

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