If tan-1a+tan-1b+tan-1c=pie,then prove that a+b+c=abc
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Answered by
34
We are asked to prove:
We will use the following concepts:
Now we can solve the question:
PART 1: Assuming ab<1
PART 2: Assuming ab>1
[tex]\tan^{-1}a+\tan^{-1}b+\tan^{-1}c = \pi \\ \\ \\ \implies \tan^{-1}a+\tan^{-1}b = \pi - \tan^{-1}c \\ \\ \\ \implies \pi + \tan^{-1}\left(\frac{a+b}{1-ab}\right) = \pi - \tan^{-1}c \\ \\ \\ \tan^{-1}\left(\frac{a+b}{1-ab}\right) = -\tan^{-1}c \\ \\ \\ \text{Taking tan on both sides} \\ \\ \\ \implies \tan \left(\tan^{-1}\left(\frac{a+b}{1-ab} \right) \right) = \tan (-\tan^{-1}c) \\ \\ \\ \implies \frac{a+b}{1-ab} = -\tan(\tan^{-1}c) \\ \\ \\ \implies \frac{a+b}{1-ab} = -c \\ \\ \\ \implies a+b = -c(1-ab) \\ \\ \\ \implies a+b = -c+abc \\ \\ \\ \implies \boxed{a+b+c=abc} \\ \\ \\ \text{Hence Proved}[/tex]
Hope it helps
Purva
Brainly Community
We will use the following concepts:
Now we can solve the question:
PART 1: Assuming ab<1
PART 2: Assuming ab>1
[tex]\tan^{-1}a+\tan^{-1}b+\tan^{-1}c = \pi \\ \\ \\ \implies \tan^{-1}a+\tan^{-1}b = \pi - \tan^{-1}c \\ \\ \\ \implies \pi + \tan^{-1}\left(\frac{a+b}{1-ab}\right) = \pi - \tan^{-1}c \\ \\ \\ \tan^{-1}\left(\frac{a+b}{1-ab}\right) = -\tan^{-1}c \\ \\ \\ \text{Taking tan on both sides} \\ \\ \\ \implies \tan \left(\tan^{-1}\left(\frac{a+b}{1-ab} \right) \right) = \tan (-\tan^{-1}c) \\ \\ \\ \implies \frac{a+b}{1-ab} = -\tan(\tan^{-1}c) \\ \\ \\ \implies \frac{a+b}{1-ab} = -c \\ \\ \\ \implies a+b = -c(1-ab) \\ \\ \\ \implies a+b = -c+abc \\ \\ \\ \implies \boxed{a+b+c=abc} \\ \\ \\ \text{Hence Proved}[/tex]
Hope it helps
Purva
Brainly Community
QGP:
Okay I accept that the condition is a.b < 1
Answered by
12
for solution refer attachment
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