Question
Download Solution PDFIf α and β are positive angles such that \(α + β = \dfrac{\pi}{4}\), then what is (1 + tan α) (1 + tan β) equal to?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
\(\rm \tan (α + β) = \dfrac {tan \alpha + tan \beta }{1 -tan \alpha \;tan \beta }\)
Calculations:
Given, α and β are positive angles such that \(α + β = \dfrac{\pi}{4}\),
⇒\(\rm \tan (α + β) = \tan (\dfrac{\pi}{4})\)
⇒\(\rm \dfrac {tan \alpha + tan \beta }{1 -tan \alpha \;tan \beta } = 1\)
⇒\(\rm {tan \alpha + tan \beta }= 1 -tan \alpha \;tan \beta \)
⇒\(\rm {tan \alpha +tan \alpha \;tan \beta + tan \beta } - 1 = 0 \)
⇒\(\rm {tan \alpha +tan \alpha \;tan \beta + tan \beta } +1-2= 0 \)
⇒\(\rm {tan \alpha +tan \alpha \;tan \beta + tan \beta } +1=2\)
⇒\(\rm tan \alpha (1 +tan \beta ) + (1+tan \beta)=2\)
⇒\(\rm (1+ tan \alpha )(1 +tan \beta )=2\)
Hence, If α and β are positive angles such that \(α + β = \dfrac{\pi}{4}\), then (1 + tan α) (1 + tan β) equal to 2.
Last updated on May 30, 2025
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