cos(2x) is equal to several different expressions, all derived from trigonometric identities. The most common form is cos²(x) - sin²(x). However, it can also be expressed in terms of just cosine or just sine.
Different Forms of cos(2x)
Here's a breakdown of the different equivalent expressions for cos(2x):
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cos(2x) = cos²(x) - sin²(x) This is the fundamental double-angle formula for cosine.
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cos(2x) = 2cos²(x) - 1 This form is derived from the previous one using the Pythagorean identity: sin²(x) + cos²(x) = 1. Substituting sin²(x) = 1 - cos²(x) into the first equation gives us this result.
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cos(2x) = 1 - 2sin²(x) Similarly, this form is also derived from cos²(x) - sin²(x) and the Pythagorean identity. This time, we substitute cos²(x) = 1 - sin²(x).
Summary
In summary, the double-angle formula for cosine, cos(2x), can be represented in three different but equivalent ways:
- cos²(x) - sin²(x)
- 2cos²(x) - 1
- 1 - 2sin²(x)
Each form is useful depending on the context of the problem you are trying to solve.
