"If α+β=π4 prove that (1+tan⁡α)(1+tan⁡β)=2" was answered by students like you

misangela4gi

Answered question

2022-04-27

α + β = \frac{π}{4}

prove that

(1 + \tan α) (1 + \tan β) = 2

Zemmiq34

Beginner2022-04-28Added 11 answers

from where OP left his step:

1 = \frac{\tan α + \tan β}{1 - \tan α \tan β}

\Rightarrow 1 - \tan α \tan β = \tan α + \tan β

\Rightarrow \tan α + \tan β + \tan α \tan β = 1

add 1 to both sides

\Rightarrow \tan α + \tan β + \tan α \tan β + 1 = 2

\Rightarrow 1 + \tan α + \tan β + \tan α \tan β = 2

factor the above equation

\Rightarrow 1 (1 + \tan α) + \tan β (1 + \tan α) = 2

\Rightarrow (1 + \tan α) (1 + \tan β) = 2

Hence proven.

2sze1c1se3nh

Beginner2022-04-29Added 17 answers

Here's another way to do it:

β = \frac{π}{4} - α \Rightarrow (1 + \tan β)

= (1 + \tan (\frac{π}{4} - α)) = (1 + \frac{1 - \tan α}{1 + \tan α}) = (\frac{2}{1 + \tan α})

Thus,

(1 + \tan β) = \frac{2}{1 + \tan α}

\Rightarrow (1 + \tan α) (1 + \tan β) = 2

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