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Apply a corollary to show an identity

  1. Feb 21, 2015 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant 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.

    3. 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.
  2. jcsd
  3. Feb 21, 2015 #2


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    Staff: 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.
  4. Feb 21, 2015 #3
    Yeah, I guess I'll simply use the Angle-Sum and -Difference Identities.
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