Question

If $cosec\, \theta - \cot \theta = 2017$, then quadrant in which $\theta$ lies is

• A. I
• B. IV
• C. III
• D. II

Answer 1

Answer: (D)

II

Answer 2

We have,
$\operatorname{cosec} \theta-\cot \theta=2017 \ldots(i)$
$\therefore \operatorname{cosec} \theta+\cot \theta=\frac{1}{2017} \ldots(ii)$
$\left[\begin{array}{l}\because \operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1 \\\ \Rightarrow \operatorname{cosec} \theta-\cot \theta=\frac{1}{\operatorname{cosec} \theta+\cot \theta}\end{array}\right]$
Adding Eqs. (i) and (ii),
we get $2 \operatorname{cosec} \theta=2017+\frac{1}{2017}$
$\Rightarrow \operatorname{cosec} \theta=\frac{1}{2}\left[2017+\frac{1}{2017}\right]>0$
$\theta$ lie in Ist or IInd quadrant.
Subtracting E (i) from E (ii), we get
$2 \cot \theta=\frac{1}{2017}-2017$
$\cot \theta=\frac{1}{2}\left(\frac{1}{2017}-2017\right) < 0$
$\therefore \theta$ lie in IInd and IIIrd quadrant.
Hence, $\theta$ lies in IInd quadrant.