Question

If $\sin x+\cos x=\frac{1}{5},$ then sum of the possible value(s) of $|12\tan x|$ is equal to

• A. $16$
• B. $9$
• C. $81$
• D. $25$

Answer 1

The correct option is D $25$

$\sin x+\cos x=\frac{1}{5}$
$\Rightarrow (\sin x+\cos x{)}^2=\frac{1}{25}$
$\Rightarrow 1+\sin 2x=\frac{1}{25}$
$\Rightarrow \frac{2\tan x}{1+{\tan }^{2}x}=-\frac{24}{25}$
$\Rightarrow 12{\tan }^{2}x+25\tan x+12=0$
$\Rightarrow (4\tan x+3)(3\tan x+4)=0$
$\Rightarrow \tan x=\frac{-3}{4}$ or $\tan x=\frac{-4}{3}$
$\Rightarrow |12\tan x|=9$ or $16$
Required sum $=9+16=25$