# Apply a corollary to show an identity

## 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
Mentor
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.

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.