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  4. If 3 sin x+4 cos x=5, then 6 tan (x/2)-9 tan 2 (x/2) is equal to

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

EAMCETEAMCET 2012

Solution:

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