Proof Of Trigonometric Identity Cos(θ)/(1 - Tan(θ)) + Sin(θ)/(1 - Cot(θ)) = Cos(θ) + Sin(θ)
This article provides a detailed proof of the trigonometric identity:
cos(θ) / (1 - tan(θ)) + sin(θ) / (1 - cot(θ)) = cos(θ) + sin(θ)
We will explore the steps involved in simplifying the left-hand side (LHS) of the equation to arrive at the right-hand side (RHS), thereby verifying the identity. This exploration will not only validate the formula but also enhance understanding of fundamental trigonometric relationships and algebraic manipulation. Understanding and proving trigonometric identities like this one is crucial for success in various fields, including physics, engineering, and computer graphics. This identity, in particular, elegantly showcases the interplay between sine, cosine, tangent, and cotangent functions, offering a valuable learning opportunity for students and enthusiasts alike.
Introduction to Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables for which the functions are defined. They are fundamental tools in trigonometry and are used to simplify expressions, solve equations, and understand the relationships between different trigonometric functions. Mastering these identities is essential for anyone studying mathematics, physics, or engineering. This particular identity, cos(θ) / (1 - tan(θ)) + sin(θ) / (1 - cot(θ)) = cos(θ) + sin(θ), is a great example of how seemingly complex trigonometric expressions can be simplified using fundamental identities and algebraic manipulations. The beauty of this identity lies in its symmetry and the way it connects different trigonometric functions. Understanding how to manipulate these functions and prove such identities is a key skill in mathematical problem-solving. We will break down the proof step-by-step, explaining the rationale behind each step and highlighting the key trigonometric identities used. This approach will help you not only understand this specific identity but also develop a more general understanding of how to approach trigonometric proofs. The ability to prove trigonometric identities is a valuable skill that will serve you well in your mathematical journey. It fosters a deeper understanding of the relationships between trigonometric functions and strengthens your algebraic manipulation skills.
Proof of the Identity
To prove the identity, we will start with the left-hand side (LHS) of the equation and manipulate it until it matches the right-hand side (RHS). This involves using various trigonometric identities and algebraic techniques. Our main goal is to rewrite the LHS in terms of sine and cosine, which will allow us to simplify the expression and eventually arrive at the RHS. The strategic use of identities like tan(θ) = sin(θ)/cos(θ) and cot(θ) = cos(θ)/sin(θ) will be crucial in this process. Moreover, careful attention to algebraic manipulation, such as finding common denominators and simplifying fractions, is key to successfully proving the identity. By meticulously following each step and understanding the underlying principles, we can demonstrate the validity of this trigonometric relationship. This process not only proves the identity but also reinforces our understanding of the fundamental connections between trigonometric functions. The satisfaction of proving an identity comes from seeing the initial complex expression transform into a simpler, more elegant form. This proof serves as an excellent example of how mathematical tools can be used to unravel intricate relationships and reveal underlying simplicity.
Step 1: Express tan(θ) and cot(θ) in terms of sin(θ) and cos(θ)
We begin by expressing tan(θ) and cot(θ) in terms of sin(θ) and cos(θ). Recall the fundamental trigonometric identities:
- tan(θ) = sin(θ) / cos(θ)
- cot(θ) = cos(θ) / sin(θ)
Substituting these identities into the LHS of the equation, we get:
cos(θ) / (1 - sin(θ) / cos(θ)) + sin(θ) / (1 - cos(θ) / sin(θ))
This step is crucial because it allows us to rewrite the expression entirely in terms of sine and cosine, which are the fundamental trigonometric functions. By expressing everything in terms of sine and cosine, we can leverage the relationships between these functions to simplify the expression further. This is a common strategy in proving trigonometric identities: to reduce the expression to its most basic components. The ability to recognize and apply these fundamental identities is essential for success in trigonometry. This substitution lays the groundwork for the subsequent steps, where we will manipulate the fractions and simplify the expression. Understanding the rationale behind this step is important for developing a general approach to proving trigonometric identities. It highlights the importance of identifying the core functions and expressing other functions in terms of them.
Step 2: Simplify the denominators
Next, we simplify the denominators of the fractions. To do this, we find a common denominator for each fraction:
cos(θ) / ((cos(θ) - sin(θ)) / cos(θ)) + sin(θ) / ((sin(θ) - cos(θ)) / sin(θ))
This step involves finding a common denominator within each fraction's denominator. By doing so, we prepare the fractions for further simplification. This is a standard algebraic technique that is widely used in manipulating complex fractions. The key here is to treat the expressions (1 - sin(θ) / cos(θ)) and (1 - cos(θ) / sin(θ)) as separate fractions and find their respective common denominators. This allows us to combine the terms in the denominator into a single fraction. Understanding this process is crucial for simplifying complex expressions and making them easier to work with. This step demonstrates the importance of algebraic manipulation in trigonometric proofs. By carefully applying algebraic rules, we can transform the expression into a more manageable form. The next step will involve dividing by these complex fractions, which will further simplify the expression.
Step 3: Divide by the fractions
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the expression as:
cos(θ) * (cos(θ) / (cos(θ) - sin(θ))) + sin(θ) * (sin(θ) / (sin(θ) - cos(θ)))
This simplifies to:
cos²(θ) / (cos(θ) - sin(θ)) + sin²(θ) / (sin(θ) - cos(θ))
This step utilizes the fundamental principle of dividing by a fraction, which is equivalent to multiplying by its reciprocal. This transformation significantly simplifies the expression by removing the complex fractions in the denominators. The ability to apply this principle fluently is crucial for manipulating fractions and simplifying algebraic expressions. By multiplying by the reciprocals, we convert the division operation into multiplication, which is often easier to handle. The resulting expression, cos²(θ) / (cos(θ) - sin(θ)) + sin²(θ) / (sin(θ) - cos(θ)), is now in a form that allows us to combine the two fractions. This step highlights the power of algebraic manipulation in simplifying trigonometric expressions. By applying a simple rule of fraction division, we have made significant progress in proving the identity. The next step will involve finding a common denominator for the two resulting fractions, which will allow us to combine them and further simplify the expression.
Step 4: Find a common denominator
To combine the two fractions, we need a common denominator. Notice that the denominators are (cos(θ) - sin(θ)) and (sin(θ) - cos(θ)). They are almost the same, except for the sign. We can make them the same by multiplying the second fraction by -1/-1:
cos²(θ) / (cos(θ) - sin(θ)) - sin²(θ) / (cos(θ) - sin(θ))
Now we have a common denominator, (cos(θ) - sin(θ)). This step is a clever maneuver that allows us to combine the two fractions. By multiplying the second fraction by -1/-1, we effectively change the sign of the denominator, making it identical to the denominator of the first fraction. This technique is particularly useful when dealing with expressions that have denominators that are opposites of each other. Recognizing this pattern and applying this technique can significantly simplify the process of finding a common denominator. This step demonstrates the importance of observing the structure of the expression and identifying opportunities for simplification. By carefully manipulating the signs, we have created a common denominator without altering the value of the expression. The next step will involve combining the fractions and simplifying the numerator.
Step 5: Combine the fractions
Now that we have a common denominator, we can combine the fractions:
(cos²(θ) - sin²(θ)) / (cos(θ) - sin(θ))
This step is a direct consequence of having a common denominator. By combining the fractions, we simplify the expression into a single fraction, making it easier to manipulate. This is a fundamental step in simplifying expressions involving fractions, and it is often a crucial step in proving identities. The numerator now contains a difference of squares, which we can factorize in the next step. This simplification highlights the importance of finding a common denominator when adding or subtracting fractions. It allows us to consolidate the terms and reveal underlying patterns in the expression. The resulting fraction is now in a form that allows us to apply further algebraic techniques, such as factorization, to simplify the expression further. This step demonstrates the power of combining fractions to reveal hidden structures and pave the way for further simplification.
Step 6: Factor the numerator
The numerator is a difference of squares, which can be factored as:
(cos(θ) - sin(θ))(cos(θ) + sin(θ)) / (cos(θ) - sin(θ))
This step utilizes the difference of squares factorization, a fundamental algebraic identity. Recognizing this pattern in the numerator allows us to factor the expression and potentially simplify it further. The difference of squares factorization, a² - b² = (a - b)(a + b), is a powerful tool in algebraic manipulation and is frequently used in trigonometric proofs. By factoring the numerator, we reveal a common factor with the denominator, which we can cancel in the next step. This step demonstrates the importance of recognizing algebraic patterns in trigonometric expressions. The ability to identify and apply algebraic identities like the difference of squares is crucial for simplifying complex expressions and proving identities. The factorization step transforms the numerator into a product of two terms, one of which matches the denominator, setting the stage for cancellation and further simplification.
Step 7: Cancel the common factor
We can now cancel the common factor (cos(θ) - sin(θ)) from the numerator and the denominator:
cos(θ) + sin(θ)
This step is a crucial simplification, as it eliminates the complex fraction and leaves us with a much simpler expression. Cancelling common factors is a fundamental algebraic technique that is widely used in simplifying expressions. The ability to identify and cancel common factors is essential for reducing complex expressions to their simplest forms. By cancelling the factor (cos(θ) - sin(θ)), we have significantly simplified the expression and brought it closer to the desired result. This step demonstrates the power of factorization in simplifying expressions. By factoring the numerator and identifying a common factor with the denominator, we were able to cancel the factor and simplify the expression considerably. The resulting expression, cos(θ) + sin(θ), is exactly the right-hand side of the original identity, completing the proof.
Conclusion
We have successfully shown that:
cos(θ) / (1 - tan(θ)) + sin(θ) / (1 - cot(θ)) = cos(θ) + sin(θ)
by manipulating the left-hand side of the equation using trigonometric identities and algebraic techniques. This proof demonstrates the power of trigonometric identities and algebraic manipulation in simplifying complex expressions. The ability to prove trigonometric identities is a valuable skill in mathematics, and this example showcases the key steps involved in such proofs. By starting with the LHS, expressing the trigonometric functions in terms of sine and cosine, simplifying the fractions, factoring, and cancelling common factors, we were able to arrive at the RHS, thus proving the identity. This proof not only validates the identity but also reinforces our understanding of the fundamental relationships between trigonometric functions. The step-by-step approach taken in this proof provides a clear roadmap for tackling similar trigonometric problems. Understanding the rationale behind each step and the application of key identities is crucial for developing a strong foundation in trigonometry. This identity serves as a great example of how seemingly complex trigonometric expressions can be simplified using a combination of trigonometric identities and algebraic techniques. Mastering these techniques is essential for success in various fields, including mathematics, physics, and engineering.
