CIRCULAR AND HYPERBOLIC FUNCTIONS 
223 
4. Since iS involves only odd powers of x, and 0 only even 
powers, 
sin x is an odd, cos x is an even function. 
5. Since S and 0 are power series which converge for every x, 
they have derivatives of every order. In particular 
dS a? x* z 6 n 
= 2l + IT“«T + - =c - 
_ _ x x> — — S 
dx~ 1 + 3T 5'.r. 
Hence 
dC 
dx 
d sin x 
dx 
COS X 
d cos x 
dx 
= — sin X. 
(3 
6. To get the addition theorem, let an index as x, y attached to 
S, C indicate the variable which occurs in the series. Then 
S x O u 
x 3 xy 2 \^ _|_ ( ^ _l_ xZ y 2 x y' 
+ 
+ 
3 ! ' 2 \J\5\ 3 ! 2 ! 1 ! 4 ! 
6 
’ X 7 xP y 2 X 3 X 4 xy' 
,7Ì + 5T2! + 3~!4T + Ì37 
+ 
+ 
^5 ^3^2 
5! 3!2! 4!1! 
Adding, 
V7! 5! 2! 3 ! 4 ! 1 ! 6 ! y 
— X + y — 
3! 
+ ( 1 )x £ y+[ i )xy*+ U 
+ ~ + (Jj^y + fy x Y + + (^J x y i + y* J + 
_ X + y (»4- y) 3 (X 4- 5) 5 
1! 
3! 
5! 
J x+y’ 
Thus for every x, y 
sin (x 4- y') — sin x cos y + cos x sin y. 
In the same way we find the addition formula for cos».