Trigonometric Identity: Derivation and Application of cos^2x - sin^2x cos2x

Trigonometry is a vital branch of mathematics, essential in understanding and solving a wide range of problems in physics, engineering, and other scientific fields. One of the fundamental identities in trigonometry that is widely utilized is the double angle identity, specifically the formula for cos^2x - sin^2x. This article will delve into the derivation of this identity and explore its applications.

Derivation of cos^2x - sin^2x cos2x

Let's begin by understanding how to derive and simplify the expression cos^2x - sin^2x.

Step 1: Using the Double Angle Formula

The double angle formula for cosine is:

Double Angle Formula: cos(A B) cosA cosB - sinA sinB

By substituting A x and B x in the formula, we get:

cos(x x) cosx cosx - sinx sinx

This simplifies to:

cos2x cos^2x - sin^2x

Step 2: Using the Pythagorean Identity

Another useful identity in trigonometry is the Pythagorean Identity, which states:

Pythagorean Identity: cos^2x sin^2x 1

Using this, we can express cos^2x as:

cos^2x 1 - sin^2x

Substituting this into the expression cos^2x - sin^2x, we get:

cos^2x - sin^2x (1 - sin^2x) - sin^2x

This simplifies to:

cos^2x - sin^2x 1 - 2sin^2x

Alternatively, we can express sin^2x as 1 - cos^2x and substitute back to get:

cos^2x - sin^2x cos^2x - (1 - cos^2x)

This simplifies to:

cos^2x - sin^2x 2cos^2x - 1

Application of cos^2x - sin^2x cos2x

The identity cos^2x - sin^2x cos2x is a powerful tool in solving trigonometric equations and simplifying expressions. Here are a few examples of its application:

Example 1: Simplifying an Expression For instance, if we have an expression such as:

(cos^2x - sin^2x) / cosx

Using the identity, we can rewrite this as:

(cos2x) / cosx

This further simplifies to:

2cosx - 1 / cosx

Example 2: Solving Trigonometric Equations Consider the equation:

cos^2x - sin^2x 0

Using the identity, this can be rewritten as:

cos2x 0

Solving for x in the range 0 to 2π, the solutions are:

x π/4, 3π/4, 5π/4, 7π/4

Conclusion

The identity cos^2x - sin^2x cos2x is not only a fundamental trigonometric identity but also an essential tool for simplifying and solving complex trigonometric expressions and equations. Understanding and memorizing this identity can significantly aid in various mathematical and scientific applications.

Further Reading

To explore more trigonometric identities and their applications, you may refer to the following resources:

Trigonometric Formulas and Identities Applications of Trigonometry in Real Life Advanced Trigonometry Concepts