Plane trigonometry, by S.L. Loney.

INVERSE CIRCULAR FUNCTIONS.

275

Let 4a= 4 +tan-1 x.
4
'120 —=tn +tan1 1 — x
^ tan( ) ltan-' x (Art. 100).
239'
Hence 4 tan-1 - - tan-1
5 239 4
Ex. 5. Prove that
a+bj
tan-1 a + tan-1 b = tan-~1 - -b
1-ab
Let tan- a = a, so that tana = a.
Let tan-1b=/, so that tan 3=b.
Also let tan- 1 (+b- b=, so that tan y=1-ab '
We have then to prove that
a+p=7y.
~~~Now t tana+tan3 - a+b
Now tan (a +) = 1- tan a tan 1 - ab
so that the relation is proved.
The above relation is merely the formula
tan x + tan y
tan (x+y) 1 - tan x tan y'
expressed in inverse notation.
For put tan x = a, so that x= tan- a,
and tan y = b, so that y =tan-l b.
Then tan (x +y) =_ ab
a+b. x+y=tan-ll-b '
1-ab
18-2