Expand the Trigonometric Expression cos(x)

Problem

cos(x)

Solution

1. Identify the expression. The expression cos(x) is a basic trigonometric function.

2. Determine if expansion is possible. Since the argument x is a single variable and not a sum, difference, or multiple (like 2*x or x+y, there are no standard trigonometric identities (such as the sum-of-angles or double-angle formulas) to apply for further expansion.

3. Represent using alternative forms. While it cannot be expanded in terms of other trigonometric functions of x it can be represented as an infinite power series (Taylor series) centered at 0

4. Apply the Taylor series formula for cosine. The expansion is given by the sum of even powers with alternating signs.

Final Answer

cos(x)=(∑_n=0^∞)(((−1)n*x(2*n))/((2*n)!))=1−(x2)/(2!)+(x4)/(4!)−(x6)/(6!)+…

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