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Question
If α + β = `pi/4`, then the value of (1 + tan α)(1 + tan β) is ______.
Options
1
2
–2
Not defined
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Solution
If α + β = `pi/4`, then the value of (1 + tanα)(1 + tanβ) is 2.
Explanation:
Given that: α + β = `pi/4`
⇒ `(tanalpha + tanbeta)/(1 - tanalpha tanbeta)` = 1
⇒ tanα + tanβ = 1 – tanα tanβ
⇒ tanα + tanβ + tanα tanβ = 1
⇒ 1 + tanα + tanβ + tanα tanβ = 1 + 1
⇒ 1(1 + tanα) + tanβ(1 + tanα) = 2
⇒ (1 + tanα)(1 + tanβ) = 2
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