If $\cot \alpha=1$ and $\sec \beta=-\frac{5}{3}$, where $\pi<\alpha<\frac{3 \pi}{2}$ and $\frac{\pi}{2}<\beta<\pi$, then the value of $\tan (\alpha+\beta)$ and the quadrant in which $\alpha+\beta$ lies, respectively are

(A) $-\frac{1}{7}$ and IV$^{\text{th}}$ quadrant

(B) 7 and $\mathrm{I}^{\mathrm{st}}$ quadrant

(C) -7 and IV$^{\text{th}}$ quadrant

(D) $\frac{1}{7}$ and $\mathrm{I}^{\mathrm{st}}$ quadrant

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Trigonometric Ratios and Identities

cot α = 1

and

sec β = - \frac{5}{3}

, where

π < α < \frac{3 π}{2}

and

\frac{π}{2} < β < π

, then the value of

tan (α + β)

and the quadrant in which

α + β

lies, respectively are

Answer: (A)

From

cot α = 1

and

π < α < \frac{3 π}{2}

(third quadrant), we have

tan α = 1

. From

sec β = - \frac{5}{3}

and

\frac{π}{2} < β < π

(second quadrant), we get

cos β = - \frac{3}{5}

, so

tan β = - \frac{4}{3}

. Using the addition formula:

tan (α + β) = \frac{t a n α + t a n β}{1 - t a n α t a n β} = \frac{1 - \frac{4}{3}}{1 + 1 \cdot ( - \frac{4}{3} )} = \frac{- \frac{1}{3}}{- \frac{1}{3}} = \frac{- 1}{7}

. Since

α \in (π, \frac{3 π}{2})

and

β \in (\frac{π}{2}, π)

, we have

α + β \in (\frac{3 π}{2}, \frac{5 π}{2})

, which places it in the fourth quadrant (after subtracting

2 π

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Single correct option