Question

If $3\sin x+4\cos x=5$, then the value of $3\tan \frac{x}{2}$ is

• A. $\frac{1\pm \sqrt{2}}{4}$
• B. $\frac{-1\pm \sqrt{2}}{4}$
• C. $\frac{-1\pm \sqrt{2}}{3}$
• D. $\frac{1}{3}$

Answer 1

The correct option is D $\frac{1}{3}$

$3\sin x+4\cos x=5$
We know that, $\sin 2\theta =\frac{2\tan \theta }{1+{\tan }^2\theta }$ and $\cos 2\theta =\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }$
$\therefore 3\left[\frac{2\tan x/2}{1+{\tan }^{2}x/2}\right]+4\left[\frac{1-{\tan }^{2}x/2}{1+{\tan }^{2}x/2}\right]=5$
$6\tan x/2+4-4{\tan }^{2}x/2=5+5{\tan }^{2}x/2$
$9{\tan }^{2}x/2-6\tan x/2+1=0$
$9{\tan }^{2}x/2-3\tan x/2-3\tan x/2+1=0$
$3\tan x/2\left[3\tan x/2-1\right]-1\left[3\tan x/2-1\right]=0$
$\left[3\tan x/2-1\right]\left[3\tan x/2-1\right]=0$
$3\tan x/2-1=0$
$3\tan x/2=\frac{1}{3}$