Question

If $\cot x=\frac{-5}{12}$ and $x$ lies in second quadrant, then choose the correct option.

• A. $\sec x=\pm \frac{13}{5}$
• B. $\tan x=\frac{12}{5}$
• C. $\sin x=-\frac{13}{12}$ and $\mathrm{cosec}x=\frac{12}{13}$
• D. $\cos x=\frac{-5}{13}$

Answer 1

Answer: (D)

Since, $\cot x=\frac{-5}{12}$, we have $\tan x=-\frac{12}{5}$
Now,
${\sec }^{2}x=1+{\tan }^{2}x$
$=1+\frac{144}{25}=\frac{169}{25}$
Hence, $\sec x=\pm \frac{13}{5}$
Since, $x$ lies in second quadrant, $\sec x$ will be negative.
Therefore, $\sec x=-\frac{13}{5}$,
which also gives $\cos x=-\frac{5}{13}$
Further, we have
and $\sin x=\tan x\cos x=\left(-\frac{12}{5}\right)\times \left(-\frac{5}{13}\right)=\frac{12}{13}$
and $\mathrm{cosec}x=\frac{1}{\sin x}=\frac{13}{12}$