Question

If 
\(cot α=1\) and \(sec β=−\frac{5}{3}\) where \(π<α<\frac{3π}{2} and \frac{π}{2}<β<π\)
, then the value of tan(α + β) and the quadrant in which α + β lies, respectively are :

• A. -1/7 and IVth quadrant
• B. 7 and Ist quadrant
• C. – 7 and IVth quadrant
• D. 1/7 and Ist quadrant

Answer 1

The Correct Option is A

The correct answer is (A) : \(-\frac{1}{7}\) and IVth quadrant
∵ cot α = 1,
\(α∈(π, \frac{3π}{2})\)
then tan α = 1 and \(sec⁡ β=−\frac{5}{3}, β∈(\frac{π}{2},π) \)
then \(tan⁡ β=−\frac{4}{3}\)
\(∴tan⁡(α+β)=\frac{tan⁡α+tan⁡β}{1−tan⁡α⋅tan⁡β}\)
\(=\frac{1−\frac{4}{3}}{1+\frac{4}{3}}\)
\(=−\frac{1}{7}\)
\(α+β∈(\frac{3π}{2},2π)\)
i.e. fourth quadrant