Question

If 2tanα + cotβ = tanβ then tan(β – α) = __________ 

(a) tanα

(b) cotα 

(c) tanβ 

(d) cotβ

Answer 1

The correct option  (d) cotβ   

Explanation:

given: 2tanα + cotβ = tanβ

∴ 2tanα = tanβ – cotβ

∴ tanα = [(tanβ – cotβ)/2]

∴ tan(β – α)   = [(tanβ – tanα)/{1 + (tanβ ∙ tanα)}]

= [{tanβ – [(tanβ)/2] + [(cotβ)/2]}/{1 + [(tan²β)/2] – (1/2)}]

= [{[(tanβ)/2] + [(cotβ)/2]}/{(1 + tan²β)/2}]

= [(tanβ + cotβ)/(1 + tan²β)]

= [{tanβ + [1/(tanβ)]}/(1 + tan²β)]

= [(1 + tan²β)/{(tanβ)(1 + tan²β)}]

= cotβ.