What are the formulas for 1 + cos(2x) and 1 – cos(2x)?

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The formulas for 1 + cos(2x) and 1 – cos(2x) can be derived using trigonometric identities.

Firstly, let’s consider 1 + cos(2x). This expression can be simplified using the half-angle identity. The cosine double angle formula states:

cos(2x) = 2cos²(x) - 1

Using this identity, we replace cos(2x) in our formula:

1 + cos(2x) = 1 + (2cos²(x) - 1) = 2cos²(x)

Next, let’s examine 1 – cos(2x). By using the same cosine double angle formula, we have:

1 - cos(2x) = 1 - (2cos²(x) - 1) = 2 - 2cos²(x) = 2(1 - cos²(x))

Since 1 - cos²(x) is equivalent to sin²(x) (by the Pythagorean identity), we can further simplify this formula:

1 - cos(2x) = 2sin²(x)

In summary, the simplified formulas are:

  • 1 + cos(2x) = 2cos²(x)
  • 1 – cos(2x) = 2sin²(x)

These identities are particularly useful in various mathematical applications, including calculus and solving trigonometric equations. By understanding these formulas, you can manipulate trigonometric expressions more effectively!

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