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How to use Parseval' identity( Plancherel?

Use an angle sum identity to verify the identity cos(2theta)=2cos^2(theta)-1?

  • Use an angle sum identity to verify the identity cos(2theta)=2cos^2(theta)-1

  • Answer:

    well you know that cos(2theta) = Cos(Theta)Cos(theta) - Sin(Theta)Sin(Theta) and you can make that Cos(2Theta) = Cos^2(theta) - Sin^2(theta) And using the fact that Cos^2 + Sin^2 = 1 you can rearrange to get Cos(2Theta) = Cos^2(theta) - (1 - Cos^2(theta) ) Cos(2theta) = Cos^2(theta) - 1 +Cos^2(theta) Cos(2theta) = 2Cos^2(theta) -1

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Given the identity: cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b), you have cos(2*t) = cos(t+t) = cos^2(t) - sin^2(t), so your identity can be written as cos^2(t) - sin^2(t) = 2*cos^2(t) - 1; Now you subtract cos^2(t), and you get: -sin^2(t) = cos^2(t) -1, and after arranging the identity : cos^2(t) + sin^2(t) = 1; which is true.

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