Verified Solution • 2025–26 Session moderate

QJEEM22Ma8886: JEE Main 2022 Mathematics - Class 11 JEEM22Ma8886 Solution: If cot = 1 and sec = , where and , then the value of and the…

If cot α\alpha = 1 and sec β\beta = −53- {5 \over 3} , where \pi < \alpha < {{3\pi } \over 2} and {\pi \over 2} < \beta < \pi , then the value of tan⁡(α+β)\tan (\alpha + \beta ) and the quadrant in which α\alpha + β\beta lies, respectively are :

Multiple Choice Options

  • A −17- {1 \over 7} and IVth quadrant
  • B 7 and Ist quadrant
  • C −- 7 and IVth quadrant
  • D 17{1 \over 7} and Ist quadrant

Step-by-Step Verified Solution

∵cot⁡α=1,α∈(π,3π2)\because \cot \alpha=1, \quad \alpha \in\left(\pi, \frac{3 \pi}{2}\right)

then tan⁡α=1\tan \alpha=1

and sec⁡β=−53,β∈(π2,π)\sec \beta=-\frac{5}{3}, \quad \beta \in\left(\frac{\pi}{2}, \pi\right)

then tan⁡β=−43\tan \beta=-\frac{4}{3}

∴tan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡α⋅tan⁡β\therefore \tan (\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \cdot \tan \beta}

amp;=1−431+43amp;=−17\begin{aligned}&=\frac{1-\frac{4}{3}}{1+\frac{4}{3}} \\\\&=-\frac{1}{7}\end{aligned}

α+β∈(3π2,2π) i.e. fourth quadrant \alpha+\beta \in\left(\frac{3 \pi}{2}, 2 \pi\right) \text { i.e. fourth quadrant }

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