Question

prove that ____tan^-1 (x-1/x-2)+tan^_1 (x+1/x+2)=90°

Answer 1

Greetings,
The answer to your question is typed below↓
___________________________________________

Given

tan^-1[ (x-1) / (x-2) ] + tan^-1[ (x+1) / (x+2) ] = 45*

First, find x.

By taking tan^-1 common we get:

tan^-1[ (x-1)/(x-2) + (x+1) / (x+2) ] = 45

Then shift tan^-1 to RHS and simultaneously find LCM for LHS 

⇒[ {(x-1)(x+2) + (x+1)(x-2)} / {(x-2)(x+2)} ] = 1                    ∵[tan 45* = 1 ]
⇒ [ (x² + 2x - x - 2 + x² - 2x + x - 2) / (x² -4) ] = 1
⇒ (2x²-4) / (x²-4) = 1 
⇒ 2x² - 4 = x² -4
⇒ x² = 0
⇒ x  = 0

Then substituting x in the given equation, we get:

⇒ tan^-1 ( 1/2) + tan^-1 (1/2) = 45*
⇒ tan^-1 (1) = 45   
= 45* = 45*  [Proved]
__________________________________________________

P.S: Enjoy;)