Put tan –1 x = α and tan –1 y = β

⇒ tan α = x and tan β = y

The given eqns. Become

a tan α + b sec α = c …(1)

and a tan β + b sec β = c …(2)

⇒ α, β are roots of a tan θ + b sec θ = c

but a tan θ + b sec θ = c

⇒ (c – a tan θ ) 2 = b 2 sec 2 θ

⇒ c 2 + a 2 tan 2 θ – 2ac tan θ = b 2 (1+ tan 2 θ )

⇒ (b 2 – a 2 ) tan 2 θ + 2 ac tan θ + (b 2 – c 2 ) = 0

∴ tan α and tan β are its roots

⇒ tan α + tan β = – … (3)

and tan α tan β = … (4)

Now, tan ( α + β ) =

=

[Using equations (1) and (2)]

=

Also tan ( α + β ) =

∴ =