Solving Trigonometric Equations Cot X = -3 And Simplifying Expressions C And D

Let's embark on a journey to unravel the trigonometric equation cot x = -3. This equation plunges us into the world of cotangent, a fundamental trigonometric function that dances with both sine and cosine. To truly grasp this equation, we must first understand the very essence of the cotangent function. The cotangent function, abbreviated as cot, is the reciprocal of the tangent function. In simpler terms, cot x is defined as the ratio of the cosine of x to the sine of x, or cos x / sin x. This relationship is pivotal to deciphering the equation cot x = -3.

Now, armed with this knowledge, we can begin to dissect the equation cot x = -3. We know that cot x is negative when either cos x is positive and sin x is negative, or when cos x is negative and sin x is positive. This crucial insight narrows our search to the second and fourth quadrants of the unit circle, where the cotangent function roams in its negative domain. To pinpoint the specific values of x that satisfy cot x = -3, we can invoke the arctangent function, also known as the inverse tangent function. The arctangent function, denoted as arctan or tan⁻¹, gracefully unveils the angle whose tangent is a given value. In this case, we seek the angle whose cotangent is -3, which means we're essentially looking for the angle whose tangent is -1/3 (since cotangent is the reciprocal of tangent).

Therefore, we can express the solution for x as x = arctan(-1/3) + nπ, where n gracefully represents any integer. This elegant expression encapsulates the infinite nature of the solutions, acknowledging the periodic behavior of trigonometric functions. The arctangent of -1/3 will yield a specific angle in the range (-π/2, π/2), and by adding multiples of π (represented by nπ), we capture all angles that share the same cotangent value of -3. These solutions gracefully reside in the second and fourth quadrants, where the cotangent function reigns negatively. We have now successfully navigated the realm of cot x = -3, understanding its profound connection to the cotangent function, the unit circle, and the arctangent function.

Let's now shift our focus to the intricate expression C = (cos³ x) / (sin³ x - cos³ x). This expression invites us to explore the interplay between trigonometric functions and algebraic manipulation. To truly decipher this expression, we must embark on a journey of simplification and transformation, guided by the fundamental identities and principles of trigonometry.

Our first step in unraveling this expression is to recognize the presence of trigonometric functions raised to the power of 3, hinting at the potential for algebraic factorization. The denominator, sin³ x - cos³ x, bears a striking resemblance to the difference of cubes factorization formula: a³ - b³ = (a - b)(a² + ab + b²). By applying this formula, we can gracefully transform the denominator into a more manageable form. Let a = sin x and b = cos x. Then, sin³ x - cos³ x elegantly becomes (sin x - cos x)(sin² x + sin x cos x + cos² x). This factorization is a pivotal step in simplifying the expression.

With the denominator now elegantly factored, we can rewrite the expression for C as C = (cos³ x) / [(sin x - cos x)(sin² x + sin x cos x + cos² x)]. A glimmer of hope emerges as we notice the presence of sin² x + cos² x within the second factor of the denominator. This familiar duo immediately reminds us of the fundamental Pythagorean identity: sin² x + cos² x = 1. By gracefully applying this identity, we can further simplify the expression.

Replacing sin² x + cos² x with 1, we have C = (cos³ x) / [(sin x - cos x)(1 + sin x cos x)]. The expression is gradually taking a more streamlined form. Now, we can explore further manipulations to potentially reveal deeper insights. One avenue to consider is dividing both the numerator and denominator by cos³ x. This maneuver aims to introduce the tangent function (tan x = sin x / cos x) and potentially simplify the expression further. Dividing both the numerator and denominator by cos³ x, we obtain C = 1 / [(sin x / cos x - 1)(1 + sin x cos x) / cos³ x)]. This step elegantly introduces tan x into the equation.

Simplifying further, we have C = 1 / [(tan x - 1)(1/cos³ x + tan x / cos² x)]. Now, we can express 1/cos³ x as sec³ x (since sec x = 1/cos x) and tan x / cos² x as tan x sec² x. The expression transforms into C = 1 / [(tan x - 1)(sec³ x + tan x sec² x)]. We have successfully navigated through algebraic manipulations and trigonometric identities, arriving at a more refined form of the expression for C. This journey exemplifies the power of strategic simplification in unraveling trigonometric mysteries. Further simplification might be possible depending on the specific context or desired form of the expression.

Let's now turn our attention to the expression D = (sin² x · cos x) / [(sin x + cos x) · (1 - sin² x)]. This expression presents a fascinating challenge, inviting us to employ our trigonometric toolkit to simplify and decipher its meaning. The presence of squared trigonometric functions and a potential Pythagorean identity in disguise suggests a clear path forward.

Our initial focus naturally gravitates towards the term 1 - sin² x in the denominator. This expression immediately triggers a familiar connection to the Pythagorean identity: sin² x + cos² x = 1. By rearranging this identity, we can elegantly express 1 - sin² x as cos² x. This substitution is a pivotal step in simplifying the expression for D.

Replacing 1 - sin² x with cos² x, we have D = (sin² x · cos x) / [(sin x + cos x) · cos² x)]. The expression now takes on a more streamlined form, revealing opportunities for further simplification. We can observe that a cos x term appears in both the numerator and the denominator, allowing us to gracefully cancel one factor of cos x. This cancellation simplifies the expression significantly.

After canceling cos x, the expression transforms into D = (sin² x) / [(sin x + cos x) · cos x)]. The expression is becoming increasingly manageable. Now, we can explore further manipulations to potentially reveal deeper insights. One possible avenue is to divide both the numerator and denominator by cos² x. This strategic move aims to introduce the tangent function (tan x = sin x / cos x) and potentially lead to further simplification.

Dividing both the numerator and denominator by cos² x, we obtain D = (sin² x / cos² x) / [(sin x + cos x) · cos x / cos² x)]. This step elegantly introduces tan² x in the numerator and simplifies the denominator. Simplifying further, we have D = tan² x / [(sin x + cos x) / cos x)]. Now, we can express (sin x + cos x) / cos x as sin x / cos x + cos x / cos x, which simplifies to tan x + 1. Therefore, the expression becomes D = tan² x / (tan x + 1). We have successfully navigated through algebraic manipulations and trigonometric identities, arriving at a more refined form of the expression for D. This journey highlights the importance of recognizing key identities and employing strategic simplification techniques. Depending on the context, further manipulation might be possible, such as expressing tan² x as (sec² x - 1) or exploring relationships with other trigonometric functions.

In this exploration, we have embarked on a fascinating journey through the world of trigonometric equations and expressions. We began by unraveling the equation cot x = -3, delving into the essence of the cotangent function and its relationship to sine, cosine, and the arctangent function. We then navigated the intricacies of the expression C = (cos³ x) / (sin³ x - cos³ x), employing algebraic factorization and the Pythagorean identity to simplify its form. Finally, we deconstructed the expression D = (sin² x · cos x) / [(sin x + cos x) · (1 - sin² x)], strategically utilizing trigonometric identities and cancellations to arrive at a more refined representation. This journey exemplifies the power of mathematical tools and techniques in transforming complex expressions into more manageable forms. The realm of trigonometry is a rich tapestry of interconnected concepts, where identities, algebraic manipulations, and strategic simplification intertwine to reveal hidden relationships and profound insights.