Math, asked by DanishKhan7073, 1 year ago

If tan-1a+tan-1b+tan-1c=pie,then prove that a+b+c=abc

Answers

Answered by QGP
34
We are asked to prove:

If \, \, \tan^{-1} a + \tan^{-1}b + \tan^{-1}c = \pi \, \, then \, \, a+b+c=abc


We will use the following concepts:

\boxed{\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)} \, \, \, if \, xy<1\\ \\ \\ \\ \boxed{\tan^{-1}x + \tan^{-1}y =\pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right)} \, \, \, if xy>1 \\ \\ \\ \boxed{\tan(\pi - \theta) = -\tan \theta} \\ \\ \\ \boxed{\tan (-\theta) = - \tan \theta}

Now we can solve the question:


PART 1: Assuming ab<1


\tan^{-1}a+\tan^{-1}b+\tan^{-1}c = \pi \\ \\ \\ \implies \tan^{-1}a+\tan^{-1}b = \pi - \tan^{-1}c \\ \\ \\ \implies \tan^{-1}\left(\frac{a+b}{1-ab}\right) = \pi - \tan^{-1}c \\ \\ \\ \text{Taking tan on both sides} \\ \\ \\ \implies \tan \left(\tan^{-1}\left(\frac{a+b}{1-ab} \right) \right) = \tan (\pi - \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}

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
QGP: But the question does not give any default conditions about a, b, and c. I kind of assumed that the conditions were valid for the formula.
QGP: Okay. Even if you take a.b > 1, you will have π - arctan (the stuff) on the LHS. The final answer will come out to be same
QGP: I probably don't have an edit option right now. I will request some Mod for an edit option, and will extend my answer tomorrow.
QGP: Edited Now.
Answered by ignitedlearner
12
for solution refer attachment
Attachments:

QGP: Brother/Sister, If arctan a = x, you cannot take a = tan x
QGP: Okay, well on another thought, you can.
QGP: Yes I get it. I was thinking of something else. Anyways, this solution seems to be perfect.
QGP: Thanks to you too. :)
QGP: [You can call me a Brother]
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