If cot α =1/2 ,sec β =-5/3 ,where π < α < 3π/2 and π/2 < β < π. Find value of tan (α + β).

Question

If cot α = \(\frac{1}{2}\),sec β = \(\frac{-5}{3}\),where π < α < \(\frac{3\pi}{2}\) and \(\frac{\pi}{2}\) < β < π. Find value of tan (α + β).

Answer 1

Given,

sec β = \(\frac{-5}{3}\)

sec²β = \(\frac{25}{9}\)

tan²β = \(\frac{25}{9}\)-1 = \(\frac{16}{9}\)

tan β = ±\(\frac{4}{3}\)

tan β = \(\frac{-4}{3}\)   (\(\frac{\pi}{2}\)< β < π)

And,  cot α = \(\frac{1}{2}\)

∴ tan α = 2,  (∵ \(\frac{\pi}{2}\)< α <\(\frac{3\pi}{2}\))

Now,

tan (α + β) = \(\frac{tanα+tanβ}{1-tanα\,tanβ}\)

\(= \frac{2+(\frac{-4}{3})}{1-2(\frac{-4}{3})}\)

= \(\frac{6-4}{3+8}\)

= \(\frac{2}{11}\)