Simplifying Trigonometric Expressions: Understanding Cos2x/2cosx-1
In trigonometry, simplifying expressions is a fundamental skill that helps in solving complex problems and understanding the underlying relationships between trigonometric functions. One such expression that often appears in advanced trigonometric equations is Cos2x/2cosx-1. Let's delve into simplifying this expression step by step, using trigonometric identities to make it more manageable.
Understanding the Expression: Cos2x/2cosx-1
The expression Cos2x/2cosx-1 may initially seem daunting, but by applying some basic trigonometric identities, we can simplify it. The goal is to break down the expression into simpler, more familiar components that are easier to work with.
Using Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions which are true for every value of the occurring variables. There are many such identities, and they can be instrumental in simplifying complex expressions. In our case, we will use the double-angle identity for cosine, which states that:
Double-Angle Identity for Cosine
Cos2x 2cos^2x - 1
Let's substitute this identity into our expression Cos2x/2cosx-1:
(2cos^2x - 1) / (2cosx - 1)
Simplifying the Expression
Now that we have substituted the double-angle identity, let's simplify the expression step by step:
Rewrite the numerator using the double-angle identity: Cos2x 2cos^2x - 1 Substitute this into the original expression: (2cos^2x - 1) / (2cosx - 1)Next, we can factor the numerator:
(2cosx - 1 cosx - 1) / (2cosx - 1)
Final Simplified Expression
After factoring, we can see that the numerator and the denominator share a common factor:
(2cosx - 1)(1 cosx) / (2cosx - 1)
By canceling out the common factor (2cosx - 1) from the numerator and the denominator, we get:
1 cosx
Therefore, the simplified form of the expression Cos2x/2cosx-1 is:
1 cosx
Conclusion
In summary, the original expression Cos2x/2cosx-1 can be simplified to 1 cosx using trigonometric identities. This process highlights the importance of recognizing and applying these identities to make complex trigonometric expressions more manageable.