Question

Prove:-

arctan(1/4)+ arctan(1/9)= (1/2)arccos(3/5)

Answer 1

Using the identity arctanA + arctanB = arctan((A+B)/(1-AB)),

arctan(1/4)+ arctan(1/9)
= arctan((1/4 + 1/9)/(1 - 1/36))
= arctan(13/35)

Now,

Using the identity
arctanx = (1/2)arccos((1-x^2)/(1+x^2))
=> arctan(13/35) = (1/2)arccos(1-(13/35)^2/1+(13/35)^2)
= (1/2)arccos(35^2-13^2/35^2+13^2)
= (1/2)arccos(1225-169/1225+169)
= (1/2)arccos(1056/1394)

I'm getting a slightly different answer. Perhaps one of the tutors could proofread my result

Answer 2

Maybe it's because the equation is false.

arctan(1/4)+arctan(1/9) = 0.3556
(1/2)arccos(3/5) = 0.4636