Q102 / 160 · TS EAMCET 2012 · Mathematics

If $3 \sin x+4 \cos x=5$, then $6 \tan \frac{x}{2}-9 \tan ^2 \frac{x}{2}$ is equal to

  • A
    $0$
  • Correct answer
    B
    $1$
  • C
    $3$
  • D
    $4$

Solution

Given, $3 \sin x+4 \cos x=5$ $\begin{aligned} & \Rightarrow 3\left(\frac{2 \tan \frac{x}{2}}{1+\tan ^2 \frac{x}{2}}\right)+4\left(\frac{1-\tan ^2 \frac{x}{2}}{1+\tan ^2 \frac{x}{2}}\right)=5 \\ & \Rightarrow 6 \tan \frac{x}{2}+4-4 \tan ^2 \frac{x}{2}=5\left(1+\tan ^2 \frac{x}{2}\right) \\ & \Rightarrow 6 \tan \frac{x}{2}-9 \tan ^2 \frac{x}{2}=1\end{aligned}$

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

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