If α+β=π4 the prove that (1+tanα)(1+tanβ)=2
Given: α+β=π4
tan(α+β)=tanπ4
tanα+tanβ1−tanαtanβ=1
⇒tanα+tanβ=1−tanαtanβ
⇒tanα+tanβ+tanαtanβ=1
⇒tanα+tanβ+tanαtanβ=1 by adding 1 to both sides.
⇒1+tanα+tanβ+tanαtanβ=1+1
⇒(1+tanα)+tanβ(1+tanβ)=1+1
∴(1+tanα)(1+tanβ)=2