What are the Formulas for 1 + cos2x and 1 - cos2x?

The exact formulas for these common trigonometric expressions are 1 + cos2x = 2cos²x and 1 - cos2x = 2sin²x. These identities are fundamental in trigonometry, simplifying complex expressions and aiding in various mathematical derivations.

Understanding These Key Trigonometric Identities

These formulas are derived directly from the double-angle identities for cosine, specifically cos2x = 2cos²x - 1 and cos2x = 1 - 2sin²x. They are crucial for transforming trigonometric expressions, especially when dealing with powers of sine and cosine.

Formula for 1 + cos2x

The formula for 1 + cos2x is given by:

1 + cos2x = 2cos²x

Derivation:
This identity can be easily understood by starting with the double-angle formula for cosine:

1. Recall one form of the double-angle identity for cosine: cos2x = 2cos²x - 1.
2. To isolate 1 + cos2x, simply add 1 to both sides of the equation:
   cos2x + 1 = 2cos²x
   Therefore, 1 + cos2x = 2cos²x.

This identity is particularly useful for reducing the power of cosine terms, transforming a term involving cos2x into a squared cosine term without a double angle.

Formula for 1 - cos2x

Similarly, the formula for 1 - cos2x is:

1 - cos2x = 2sin²x

Derivation:
This identity is derived from another form of the double-angle formula for cosine:

1. Recall another form of the double-angle identity for cosine: cos2x = 1 - 2sin²x.
2. Rearrange this equation to isolate 1 - cos2x. Subtract cos2x from both sides and add 2sin²x to both sides:
   2sin²x = 1 - cos2x
   Therefore, 1 - cos2x = 2sin²x.

This identity is invaluable for reducing the power of sine terms, converting a cos2x term into a squared sine term.

Practical Applications of These Formulas

These identities are more than just theoretical concepts; they have wide-ranging applications in various fields of mathematics and science:

• Calculus: They are frequently used in integral calculus to simplify integrands, making it easier to integrate expressions involving cos²x or sin²x. For example, to integrate cos²x, you can rewrite it as (1 + cos2x) / 2.
• Trigonometric Simplification: They help simplify complex trigonometric expressions and prove other trigonometric identities. By converting terms involving cos2x, these formulas enable the identification of various trigonometric relationships.
• Physics and Engineering: These identities are applied in areas like wave mechanics, signal processing, and electrical engineering, where oscillations and periodic functions are common.
• Solving Equations: They can be used to solve trigonometric equations by transforming them into simpler forms.

Summary of Formulas

These formulas are fundamental tools for anyone working with trigonometry, offering elegant solutions for simplifying and manipulating trigonometric expressions.