19. tan-¹ (a-b 1+ ab) + tan-¹ (b-c 1 + bc) is equal to
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Answer:
Let us assume that a=tanα, b=tanβ and c=tanγ
∴tan
−1
(
1+ab
a−b
)=tan
−1
(
1+tanαtanβ
tanα−tanβ
)
=tan
−1
(tan(α−β))
=α−β
Similarly, tan
−1
(
1+bc
b−c
)=β−γ and tan
−1
(
1+ac
c−a
)=γ−α
∴tan
−1
(
1+ab
a−b
)+tan
−1
(
1+bc
b−c
)+tan
−1
(
1+ac
c−a
)=α−β+β−γ+γ−α=
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