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Question
If tanα = `m/(m + 1)`, tanβ = `1/(2m + 1)`, then α + β is equal to ______.
Options
`pi/2`
`pi/3`
`pi/6`
`pi/4`
Solution
If tanα = `m/(m + 1)`, tanβ = `1/(2m + 1)`, then α + β is equal to `bbunderline(pi/4)`.
Explanation:
Given that tanα = `m/(m + 1)`, tanβ = `1/(2m + 1)`
tan(α + β) = `(tanalpha + tanbeta)/(1 - tanalpha tanbeta)`
= `(m/(m + 1) + 1/(2m + 1))/(1 - m/(m + 1) xx 1/(2m + 1))`
= `((2m^2 + m + m + 1)/((m + 1)(2m + 1)))/(((m + 1)(2m + 1) - m)/((m + 1)(2m + 1))`
= `(2m^2 + 2m + 1)/(2m^2 + 2m + m + 1 - m)`
= `(2m^2 + 2m + 1)/(2m^2 + 2m + 1)`
= 1
⇒ tan(α + β) = `tan pi/4`
∴ α + β = `pi/4`
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