Simplify Cos(2θ) / (cosθ + Sinθ): A Step-by-Step Guide

by Alex Johnson 55 views

Introduction

In the realm of trigonometry, simplifying expressions is a fundamental skill. This article delves into the simplification of a specific trigonometric expression: cos(2θ) / (cosθ + sinθ). We will explore the steps involved in simplifying this expression, paying close attention to the values of θ for which the expression is defined. Trigonometric simplification involves using trigonometric identities and algebraic manipulations to rewrite a trigonometric expression in a simpler form. This is a crucial skill in mathematics, particularly in calculus and physics, where complex expressions often need to be simplified for further analysis or computation. Simplifying trigonometric expressions not only makes them easier to work with but also reveals underlying relationships and structures within mathematical problems. In the specific expression we are examining, cos(2θ) / (cosθ + sinθ), we encounter a combination of double-angle formulas and basic trigonometric functions. The double-angle formula for cosine, which can be expressed in several forms, plays a key role in the simplification process. Additionally, the denominator, which is the sum of sine and cosine functions, requires careful consideration to avoid division by zero. Before diving into the simplification steps, it's important to understand the domain of the expression. We need to identify the values of θ for which the expression is defined, which means ensuring that the denominator (cosθ + sinθ) does not equal zero. By carefully considering these factors and applying the appropriate trigonometric identities, we can simplify the expression and gain a deeper understanding of its behavior.

Understanding the Domain: Where is cos(2θ) / (cosθ + sinθ) Defined?

Before we dive into simplifying the expression cos(2θ) / (cosθ + sinθ), it's crucial to understand its domain. The domain refers to the set of all possible input values (in this case, θ) for which the expression is defined. In simpler terms, we need to find out which values of θ we can plug into the expression without causing any mathematical errors. The main concern here is division by zero. A fraction is undefined when its denominator is zero. Therefore, the expression cos(2θ) / (cosθ + sinθ) is undefined when cosθ + sinθ = 0. To find these problematic values of θ, we need to solve the equation cosθ + sinθ = 0. We can rewrite this equation as sinθ = -cosθ. Dividing both sides by cosθ (assuming cosθ ≠ 0), we get tanθ = -1. The tangent function equals -1 at angles of the form θ = (3π/4) + nπ, where n is an integer. These are the values of θ that make the denominator zero and thus must be excluded from the domain. We also need to consider the cases where cosθ = 0, which we initially excluded when dividing by cosθ. cosθ = 0 at θ = (π/2) + nπ, where n is an integer. However, at these values, sinθ is either 1 or -1, so cosθ + sinθ is not zero. Thus, we don't need to exclude these values separately since they are already covered by the tanθ = -1 condition. Therefore, the expression cos(2θ) / (cosθ + sinθ) is defined for all values of θ except θ = (3π/4) + nπ, where n is any integer. Understanding the domain is crucial because it ensures that any simplification we perform is valid for all allowed values of θ. Now that we've identified the domain, we can proceed with simplifying the expression.

Simplifying the Expression: Step-by-Step

Now that we know the values of θ for which our expression is defined, let's dive into the simplification process. Our goal is to make the expression cos(2θ) / (cosθ + sinθ) as concise and manageable as possible. The first step in simplifying this expression involves using the double-angle formula for cosine. There are several forms of this formula, but the most useful one for our purpose is: cos(2θ) = cos²(θ) - sin²(θ). Substituting this into our original expression, we get: (cos²(θ) - sin²(θ)) / (cosθ + sinθ). Notice that the numerator is a difference of squares. We can factor it as: cos²(θ) - sin²(θ) = (cosθ - sinθ)(cosθ + sinθ). This factorization is a key step because it introduces a term that matches the denominator. Now our expression looks like this: [(cosθ - sinθ)(cosθ + sinθ)] / (cosθ + sinθ). We can now cancel the common factor of (cosθ + sinθ) from the numerator and the denominator, provided that cosθ + sinθ ≠ 0, which we've already established when determining the domain. After canceling the common factor, we are left with: cosθ - sinθ. This is the simplified form of the expression. It's much more compact and easier to work with than the original expression. This simplified form, cosθ - sinθ, clearly shows the relationship between the cosine and sine functions in the expression. It also allows us to easily evaluate the expression for any value of θ within its domain. By using the double-angle formula and factoring the difference of squares, we were able to transform a seemingly complex expression into a simple and elegant form. In the next section, we'll verify our simplified expression and ensure that it is equivalent to the original expression for all valid values of θ.

Verifying the Simplified Expression

After simplifying the expression cos(2θ) / (cosθ + sinθ) to cosθ - sinθ, it's important to verify that our simplified form is indeed equivalent to the original expression. This verification step helps ensure that we haven't made any errors during the simplification process. To verify the equivalence, we can use a combination of algebraic manipulation and trigonometric identities. We start with our simplified expression, cosθ - sinθ, and try to manipulate it back into the original expression, cos(2θ) / (cosθ + sinθ). To do this, we can multiply the simplified expression by (cosθ + sinθ) / (cosθ + sinθ), which is equal to 1 (as long as cosθ + sinθ ≠ 0, which we've already established): (cosθ - sinθ) * [(cosθ + sinθ) / (cosθ + sinθ)]. Multiplying the terms in the numerator, we get: [(cosθ - sinθ)(cosθ + sinθ)] / (cosθ + sinθ). The numerator is now in the form of a difference of squares, which we can rewrite as: (cos²(θ) - sin²(θ)) / (cosθ + sinθ). Now, we can use the double-angle formula for cosine, which states that cos(2θ) = cos²(θ) - sin²(θ). Substituting this into our expression, we get: cos(2θ) / (cosθ + sinθ). This is the original expression we started with. Therefore, we have successfully verified that our simplified expression, cosθ - sinθ, is equivalent to the original expression, cos(2θ) / (cosθ + sinθ), for all values of θ in the domain. Another way to verify the simplified expression is to substitute specific values of θ into both the original and simplified expressions and check if the results are the same. For example, we can try θ = 0, θ = π/4, and θ = π. However, this method only provides verification for the chosen values and doesn't guarantee equivalence for all values in the domain. The algebraic manipulation method, as we've shown, provides a more rigorous proof of equivalence. In conclusion, by multiplying our simplified expression by a clever form of 1 and using the double-angle formula, we have confidently verified that cosθ - sinθ is indeed the simplified form of cos(2θ) / (cosθ + sinθ).

Alternative Forms and Their Validity

While we've simplified the expression cos(2θ) / (cosθ + sinθ) to cosθ - sinθ, it's interesting to explore alternative forms and their validity. This can provide a deeper understanding of the expression and its behavior. One alternative form that might come to mind involves using the other double-angle formulas for cosine. Recall that cos(2θ) can also be expressed as 2cos²(θ) - 1 and 1 - 2sin²(θ). However, substituting these into the original expression doesn't lead to a straightforward simplification. Another approach is to try to rewrite cosθ - sinθ in a different form. We can multiply and divide by √2: √2 [(1/√2)cosθ - (1/√2)sinθ]. Notice that 1/√2 is both the sine and cosine of π/4. So, we can rewrite the expression as: √2 [cos(π/4)cosθ - sin(π/4)sinθ]. Using the cosine addition formula, cos(A + B) = cosAcosB - sinAsinB, we can further simplify this to: √2 cos(θ + π/4). This is an alternative form of the simplified expression. It expresses the original expression as a cosine function with a phase shift and a scaling factor. This form can be useful in certain contexts, such as analyzing the amplitude and phase of a trigonometric function. It's important to note that all these forms, cosθ - sinθ and √2 cos(θ + π/4), are equivalent to the original expression cos(2θ) / (cosθ + sinθ), but they may be more convenient to use in different situations. When working with trigonometric expressions, it's often beneficial to explore different forms and choose the one that best suits the problem at hand. This requires a solid understanding of trigonometric identities and algebraic manipulation techniques. Furthermore, it's always crucial to remember the domain of the expression and ensure that any manipulations are valid for all values within the domain. In summary, while cosθ - sinθ is the simplest form we found, the alternative form √2 cos(θ + π/4) provides a different perspective on the expression and its properties. Exploring these alternative forms enhances our understanding of trigonometric functions and their interrelationships.

Conclusion

In this article, we embarked on a journey to simplify the trigonometric expression cos(2θ) / (cosθ + sinθ). We began by understanding the domain of the expression, identifying the values of θ for which the expression is defined. This involved recognizing that the denominator, cosθ + sinθ, cannot be equal to zero. We then proceeded with the simplification process, utilizing the double-angle formula for cosine and factoring the difference of squares. This led us to the simplified form cosθ - sinθ. To ensure the validity of our simplification, we verified the simplified expression by manipulating it back into the original form using trigonometric identities and algebraic techniques. We also explored alternative forms of the simplified expression, such as √2 cos(θ + π/4), which provides a different perspective on the expression's properties. Throughout this process, we emphasized the importance of understanding trigonometric identities and algebraic manipulation techniques. These skills are essential for simplifying trigonometric expressions and solving related problems in mathematics, physics, and engineering. Simplifying trigonometric expressions not only makes them easier to work with but also reveals underlying relationships and structures within mathematical problems. The ability to simplify expressions is a valuable tool in any mathematician's arsenal. In conclusion, we successfully simplified the expression cos(2θ) / (cosθ + sinθ) to cosθ - sinθ, and we gained a deeper understanding of the expression and its behavior along the way. Remember to always consider the domain of an expression before simplifying it, and to verify your results to ensure accuracy. For further exploration of trigonometric identities and simplification techniques, you may find the resources available at Khan Academy Trigonometry helpful.