If α and β are positive angles such that \(α + β = \dfrac{\pi}{

1.	If 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» Correct Answer - Option 3 : 2 Concept: \(\rm \tan (α + β) = \dfrac {tan \alpha + tan \beta }{1 -tan \alpha \;tan \beta }\) 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.

If 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» Correct Answer - Option 3 : 2

Concept:

\(\rm \tan (α + β) = \dfrac {tan \alpha + tan \beta }{1 -tan \alpha \;tan \beta }\)

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.