Apply a corollary to show an identity

[Shackleford]

Homework Statement

Apply corollary to show that 2 sinz*sinw = cos(z-w) - cos(z+w) for any z,w ∈ ℂ

Homework Equations

2 sinz*sinw = cos(z-w) - cos(z+w) for any z,w ∈ ℂ

Corollary: Let f and g be analytic functions defined on a domain D ⊂ ℂ. Let E ⊂ D be a subset that has at least one limit point a in D. If f|_E = g|_E, then f = g in D.

The Attempt at a Solution

Of course, this trig identity holds in both the real and complex spaces. As I understand it, the notation f|_E means that f is restricted to the subset E. I'm not sure if it's as easy as simply plugging in two values from E and then claiming that the functions are equal and then extending that to the larger domain D.

 

[mfb]

Two values are not enough, the identity has to hold for every point in E in order to use the corollary - but you have the freedom to choose E as long as it satisfies the condition with the limit point.

I guess you are allowed to use the standard trigonometric identities for real numbers here.

 

[Shackleford]

Two values are not enough, the identity has to hold for every point in E in order to use the corollary.

I guess you are allowed to use the standard trigonometric identities for real numbers here.

Yeah, I guess I'll simply use the Angle-Sum and -Difference Identities.