HYPERBOLIC FUNCTIONS 57 



(c) Working with sin 6 a cos 4 a 



(2i sin a)(2 cos a) 4 = (x - J(x + 1)* 



64 sin 8 a 16 cos 4 a = (a* - ^) its - -Y 

 \ a: 2 / \ x/ 



1024 sin a cos 4 a = (a:* -4x* +6 -4-i + ^](x z -2 +i 



\ X* X*/\ X 



, gl J_ ,1 1 



8 ~~ + 



= 2 cos 10a 4 cos 8a 6 cos 6a 

 + 16 cos 4<x + 4 cos 2a 12 



and sin 6 a cos 4 a = -37^(6 2 cos 2a 8 cos 4a + 3 cos 6a 

 51Z 



+ 2 cos 8a cos 10a} 

 36. The Hyperbolic Functions. The expressions -(e? e~ a ) and 



) are spoken of as the hyperbolic sine and cosine re- 

 speed vely, of the angle a 



and symbolically sinh a = -(t? e-) 



m 



cosh a = -(e? + e-) 



& 



e e~ a 



also tanh a = - - 

 e+ e-" 



(a) Then cosh 2 a - sinh 2 a = ^{(^ + e~ a ) z - (e* - e-*)*} 



4 



= ^{e 2a + 2 + e~ 2a - c 28 



4 



= 1 



(b) 2 sinh a cosh a = 2 1 (* - ^^(e 8 + e^>) 



= sinh 2a 



