2008/010) If P = arctan 1 + 1/2.arctan 2 + 1/3.arctan 3 and Q = arctan 1 + 2 arctan 1/2 + 3 arctan 1/3, then (P - Q)/5 = :

we know arctan 1/x = pi/2 - arc tan x

so Q = arctan 1 + 2(pi/2 - arc tan 2) + 3(pi/2 - arc tan 3)
= 1 + 5pi/2 - 2 arc tan 2 - 3 arc tan 3

so P-Q = 1/2 arc tan 2 + 1/3 arctan 3 -(5pi/2 - 2 arc tan 2 - 3 arc tan 3)
= 5/2 arctan 2 + 10/3 arc tan 3 - 5pi/2

devide by 5 to get
(P-Q)/5 = 1/2 arc tan 2 + 2/3 arc tan 3 - pi/2

= 1/6( 3 arc tan 2 + 4 arc tan 3) - pi/2

3 arctan 2+ 4 arctan 3
= 3 (arc tan 2 + arc tan 3 ) + arctan 3

we know arc tan 2 + arc tan 3 = arctan ( 2+3)/(1-2*3) = arc tan (-1) = - pi/4


so (P-Q)/5 = 1/6(arctan 3 - 3 pi/4)
= 1/6 arctan 3 - pi/8