Cosine Last Year, Sine Next Year

Algebra Level 5

If \( \cos (2013 x) \) is stated in terms of a polynomial of \( \cos (x) \), the coefficient of \( \cos^{2013} (x) \) is equals to \( A \cdot 2^B \)

And if \( \sin (2015 x) \) is stated in terms of a polynomial of \( \sin (x) \), the coefficient of \( \sin^{2015} (x) \) is equals to \( C \cdot 2^D \)

Where \(A, B, C, D\) are integers, with \(A \) and \(C \) as odd numbers. What is the value of \((A+B)-(C+D)\)?

Details and assumptions:

  • As an explicit example: if \( \cos(3x) \) is stated in terms of a polynomial of \( \cos(x) \), it would be \( 4 \cos^3 (x) - 3\cos (x) \)
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