Find x if arctan(x+3)−arctan(x−3)=arctan(3/4)

yasusar0 2022-07-18 Answered
Solving inverse trig question without forming cases
Question:
Find x if arctan ( x + 3 ) arctan ( x 3 ) = arctan ( 3 / 4 )
My attempt:
I know the formula:
arctan x arctan y = arctan ( x y 1 + x y )
for both x , y > 0
If both x , y are NOT greater than 0 then we'll need to form a second case in the above formula.
I solved assuming both x + 3 and x 3 are greater than zero and got a quadratic in the end solving which I got x = ± 4
I dutifully rejected x = 4 as it invalidated my assumption. However on putting in the above equation I found that it DOES satisfy the equation.
I do not wish to form cases for positive/negative. I wish to know if there is a simpler and direct way to solve such a question. Thanks!
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Answers (1)

Vartavk
Answered 2022-07-19 Author has 11 answers
Your answer { 4 , 4 } is right.
A full solution:
Since tan ( α β ) = tan α tan β 1 + tan α tan β and tan arctan x = x for all real x, we obtain
tan ( arctan ( x + 3 ) arctan ( x 3 ) ) = tan arctan 3 4
or
x + 3 ( x 3 ) 1 + ( x + 3 ) ( x 3 ) = 3 4
or
x 2 = 16 ,
which gives x = 4 or x = 4
The checking of these numbers gets that indeed, they are roots and we are done!

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