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Products and powers of trigonometric terms involving sin, cos, sinh and cosh are combined into a sum of trigonometric terms by repeated application of the transformations
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where and are special cases of the above. The form of the result is a sum of trigonometric terms whose arguments are integral linear combinations of the original arguments.
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An important special case is when the input is a polynomial in and over a field, in which case the result is a canonical form; namely,
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where , are in the field and is bounded by the total degree of the input polynomial in and .
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