Prove The Identity: Cot⁻¹(√(1+x) - √(1-x) / √(1+x) + √(1-x)) = Π/4 + 1/2cos⁻¹x

Introduction: Unraveling the Intricacies of Inverse Trigonometric Functions

In the fascinating realm of mathematics, inverse trigonometric functions hold a special allure, offering a gateway to unraveling the relationships between angles and sides in triangles. Among these functions, the inverse cotangent, denoted as cot⁻¹(x), stands out as a valuable tool for determining the angle whose cotangent is x. This exploration delves into a captivating identity involving the inverse cotangent function: cot⁻¹(√(1+x) - √(1-x) / √(1+x) + √(1-x)) = π/4 + 1/2cos⁻¹x. This identity elegantly connects the inverse cotangent function with the inverse cosine function, revealing a profound relationship between these seemingly distinct mathematical entities. To fully grasp the essence of this identity, we will embark on a step-by-step journey, meticulously dissecting its components and employing a blend of algebraic manipulations and trigonometric insights. Our exploration will not only validate the identity but also illuminate the underlying mathematical principles that govern its structure. By the end of this analysis, you will gain a deeper appreciation for the interconnectedness of mathematical concepts and the power of mathematical reasoning.

Dissecting the Identity: A Step-by-Step Exploration

To embark on our journey of unraveling this identity, let's first break it down into its constituent parts. The left-hand side (LHS) of the equation, cot⁻¹(√(1+x) - √(1-x) / √(1+x) + √(1-x)), presents a seemingly complex expression involving square roots and the inverse cotangent function. The right-hand side (RHS), π/4 + 1/2cos⁻¹x, appears more straightforward, comprising a constant term (π/4) and half the inverse cosine of x. Our primary goal is to demonstrate that these two expressions are indeed equivalent. To achieve this, we will strategically manipulate the LHS using a series of algebraic and trigonometric techniques. A pivotal step in our approach involves employing a clever substitution. Let's introduce a new variable, x = cos(θ), where θ lies within the interval [0, π]. This substitution is motivated by the presence of the inverse cosine function on the RHS and aims to simplify the expression within the inverse cotangent function. By carefully applying this substitution and leveraging trigonometric identities, we will gradually transform the LHS into a form that closely resembles the RHS. This meticulous process will not only validate the identity but also provide a deeper understanding of the underlying mathematical relationships.

The Substitution Gambit: Unveiling the Trigonometric Transformation

Now, let's put our substitution, x = cos(θ), into action. By replacing x with cos(θ) in the LHS, we obtain a new expression: cot⁻¹(√(1+cos(θ)) - √(1-cos(θ)) / √(1+cos(θ)) + √(1-cos(θ))). This substitution might seem like a small step, but it paves the way for a significant simplification. To proceed further, we will harness the power of trigonometric identities. Recall the half-angle identities: 1 + cos(θ) = 2cos²(θ/2) and 1 - cos(θ) = 2sin²(θ/2). These identities are crucial as they allow us to express the terms under the square roots in a more manageable form. Substituting these identities into our expression, we get:

cot⁻¹(√(2cos²(θ/2)) - √(2sin²(θ/2)) / √(2cos²(θ/2)) + √(2sin²(θ/2)))

Now, we can simplify the square roots: √(2cos²(θ/2)) = √2 |cos(θ/2)| and √(2sin²(θ/2)) = √2 |sin(θ/2)|. Since θ lies in the interval [0, π], θ/2 will lie in the interval [0, π/2], where both sine and cosine are non-negative. Therefore, we can drop the absolute value signs, resulting in:

cot⁻¹(√2 cos(θ/2) - √2 sin(θ/2) / √2 cos(θ/2) + √2 sin(θ/2))

We can now factor out √2 from both the numerator and the denominator, leading to further simplification:

cot⁻¹(cos(θ/2) - sin(θ/2) / cos(θ/2) + sin(θ/2))

This transformation has brought us closer to our goal. The expression within the inverse cotangent function is now in a form that we can manipulate further using trigonometric identities.

The Tangent Transformation: Bridging Cotangent and Tangent

Our next strategic maneuver involves transforming the expression within the inverse cotangent function into a form that resembles the tangent of an angle. To achieve this, we will divide both the numerator and the denominator of the fraction by cos(θ/2). This yields:

cot⁻¹((cos(θ/2) - sin(θ/2)) / cos(θ/2) / (cos(θ/2) + sin(θ/2)) / cos(θ/2))

Simplifying this expression, we get:

cot⁻¹(1 - tan(θ/2) / 1 + tan(θ/2))

At this juncture, we invoke a crucial trigonometric identity: tan(π/4 - A) = (1 - tan(A)) / (1 + tan(A)). This identity provides a direct link between the expression we have obtained and the tangent of a difference of angles. By recognizing this connection, we can rewrite our expression as:

cot⁻¹(tan(π/4 - θ/2))

Now, we need to express the tangent in terms of cotangent to effectively utilize the inverse cotangent function. Recall the identity: cot(π/2 - A) = tan(A). Applying this identity, we can rewrite the expression as:

cot⁻¹(cot(π/2 - (π/4 - θ/2)))

Simplifying the argument of the cotangent function, we get:

cot⁻¹(cot(π/4 + θ/2))

This transformation has brought us to a pivotal point in our analysis. We have successfully expressed the LHS in terms of the inverse cotangent of the cotangent of an angle.

The Inverse Cotangent Dance: Revealing the Final Form

Now, we can utilize the fundamental property of inverse functions: cot⁻¹(cot(x)) = x, provided that x lies within the range of the inverse cotangent function, which is (0, π). In our case, the argument of the outer cotangent function is π/4 + θ/2. Since θ lies in the interval [0, π], θ/2 lies in the interval [0, π/2], and therefore, π/4 + θ/2 lies in the interval [π/4, 3π/4], which is indeed within the range of the inverse cotangent function. Thus, we can apply the property, yielding:

π/4 + θ/2

Our journey is nearing its end. We have successfully transformed the LHS into this concise expression. Now, we need to express this result in terms of the original variable, x. Recall our initial substitution: x = cos(θ). Therefore, θ = cos⁻¹(x). Substituting this back into our expression, we obtain:

π/4 + 1/2 cos⁻¹(x)

Lo and behold, this is precisely the RHS of the identity we set out to prove! We have successfully demonstrated that the LHS and RHS are indeed equivalent.

Conclusion: A Symphony of Mathematical Harmony

In this detailed exploration, we have meticulously dissected the identity cot⁻¹(√(1+x) - √(1-x) / √(1+x) + √(1-x)) = π/4 + 1/2cos⁻¹x. Through a strategic combination of algebraic manipulations, trigonometric substitutions, and the application of key trigonometric identities, we have successfully transformed the LHS into the RHS, thereby validating the identity. This journey has not only demonstrated the validity of the identity but has also illuminated the interconnectedness of mathematical concepts. The interplay between inverse trigonometric functions, algebraic manipulations, and trigonometric identities highlights the elegance and harmony within the mathematical landscape. The identity serves as a testament to the power of mathematical reasoning and the beauty of mathematical relationships. By delving into such identities, we gain a deeper appreciation for the richness and depth of mathematics, fostering a sense of wonder and curiosity that drives further exploration.

This identity has practical applications in various fields, including physics and engineering, where inverse trigonometric functions are used to solve problems involving angles and distances. Furthermore, understanding this identity enhances one's problem-solving skills and provides a valuable tool for tackling complex mathematical challenges. The ability to manipulate trigonometric expressions and recognize underlying relationships is crucial for success in advanced mathematics and related disciplines. As you continue your mathematical journey, remember that each identity, each theorem, and each concept is a piece of a larger puzzle, contributing to a more complete and nuanced understanding of the mathematical world. Embrace the challenge, explore the connections, and let the beauty of mathematics inspire your intellectual pursuits.

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