NIMCET 2017 — Mathematics PYQ

Concept:
tan² x + 1 = sec² x
$tanx=sinx/cosx$
$sin2x=2sinxcosx$

Calculation:

Here, $\tan x=\frac{-3}{4}$

Squaring and adding 1 to both the sides, we get
${\tan }^{2}x+1={\left(\frac{-3}{4}\right)}^2+1$

${\sec }^{2}x=\frac{25}{16}$

$\sec x=\pm \frac{5}{4}$

$x$ is in fourth quadrant so, $\sec x=\frac{5}{4}$ and $\cos x=\frac{4}{5}$

Now

$\frac{\sin x}{\cos x}=\frac{-3}{4}$

$\sin x=\frac{-3}{4}\times \frac{4}{5}=\frac{-3}{5}$

Now, $\sin 2x=2\sin x\cos x$

$=2\times \frac{-3}{5}\times \frac{4}{5}$

$=\frac{-24}{25}$

Hence, option (4) is correct.