Verify Trigonometric Identity -csc(t)/sin(t) + Cot(t)/tan(t) = -1

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In the realm of trigonometry, verifying identities is a fundamental skill. Trigonometric identities are equations that hold true for all values of the variables involved. Mastering the art of verifying these identities not only deepens your understanding of trigonometric functions but also equips you with the tools to solve complex problems in mathematics, physics, and engineering. This article provides a comprehensive guide to verifying the identity: -csc(t)/sin(t) + cot(t)/tan(t) = -1. We will embark on a step-by-step journey, breaking down each transformation and explaining the underlying trigonometric principles. By the end of this guide, you will have a clear understanding of how to manipulate trigonometric expressions and confidently verify identities.

The core concept behind verifying trigonometric identities lies in the manipulation of one side of the equation until it matches the other side. We achieve this through a combination of fundamental trigonometric identities, algebraic techniques, and a strategic approach. It's like solving a puzzle, where each step brings you closer to the final solution. The key is to choose the more complex side of the equation and simplify it until it becomes identical to the simpler side. This process often involves expressing trigonometric functions in terms of sine and cosine, applying Pythagorean identities, and using algebraic manipulations such as factoring or combining fractions.

Before diving into the specifics of the given identity, let's briefly review some essential trigonometric identities. These identities serve as the building blocks for verifying more complex identities. The reciprocal identities, for instance, define the relationships between the six basic trigonometric functions: cosecant (csc), secant (sec), cotangent (cot), sine (sin), cosine (cos), and tangent (tan). The Pythagorean identities, derived from the Pythagorean theorem, provide fundamental relationships between sine, cosine, and tangent. The quotient identities express tangent and cotangent in terms of sine and cosine. Understanding and memorizing these fundamental identities is crucial for successfully navigating the world of trigonometric identity verification.

In this section, we will provide a detailed, step-by-step solution to verify the identity -csc(t)/sin(t) + cot(t)/tan(t) = -1. We will begin by selecting the left-hand side of the equation, which appears to be more complex, and then systematically transform it until it matches the right-hand side. Each step will be clearly explained, highlighting the trigonometric identities and algebraic manipulations employed.

  1. Starting with the Left-Hand Side:

    Our starting point is the left-hand side of the equation:

    -csc(t)/sin(t) + cot(t)/tan(t)

    The goal is to manipulate this expression using known trigonometric identities and algebraic techniques until it simplifies to -1, which is the right-hand side of the equation. The complexity of this side provides ample opportunity to apply our trigonometric knowledge and demonstrate the verification process.

  2. Expressing in Terms of Sine and Cosine:

    The first strategic move is to express all trigonometric functions in terms of sine and cosine. This simplifies the expression and makes it easier to apply other identities. Recall the following reciprocal and quotient identities:

    • csc(t) = 1/sin(t)
    • cot(t) = cos(t)/sin(t)
    • tan(t) = sin(t)/cos(t)

    Substituting these identities into our expression, we get:

    -(1/sin(t))/sin(t) + (cos(t)/sin(t))/(sin(t)/cos(t))

    This substitution is a crucial step in simplifying the expression. By converting all functions to sine and cosine, we create a common ground for further manipulation. This technique is frequently used in verifying trigonometric identities and is a valuable tool to master.

  3. Simplifying the Expression:

    Next, we simplify the expression by performing the divisions:

    -1/sinĀ²(t) + cosĀ²(t)/sinĀ²(t)

    Here, we have simplified the fractions by dividing the numerators and denominators. This step brings us closer to our goal by reducing the complexity of the expression and revealing opportunities for further simplification. We now have two terms with a common denominator, which allows us to combine them into a single fraction.

  4. Combining Fractions:

    Now, we combine the two fractions since they have a common denominator:

    (-1 + cosĀ²(t))/sinĀ²(t)

    Combining fractions is a fundamental algebraic technique that is essential for simplifying expressions. By combining the two terms into a single fraction, we have created an opportunity to apply the Pythagorean identity, which relates sine and cosine. This step demonstrates the interplay between algebra and trigonometry in the verification process.

  5. Applying the Pythagorean Identity:

    Recall the Pythagorean identity:

    sinĀ²(t) + cosĀ²(t) = 1

    Rearranging this identity, we get:

    cosĀ²(t) - 1 = -sinĀ²(t)

    Substituting this into our expression, we have:

    -sinĀ²(t)/sinĀ²(t)

    The Pythagorean identity is a cornerstone of trigonometry and is frequently used in simplifying trigonometric expressions. Recognizing the opportunity to apply this identity is a key skill in verifying trigonometric identities. The substitution here significantly simplifies the expression, bringing us closer to the final result.

  6. Final Simplification:

    Finally, we simplify the fraction:

    -sinĀ²(t)/sinĀ²(t) = -1

    Thus, we have shown that the left-hand side of the equation is equal to -1, which is the right-hand side. This final step completes the verification process, demonstrating that the given identity is indeed true. The simplification highlights the elegance and power of trigonometric identities in simplifying complex expressions.

In conclusion, we have successfully verified the trigonometric identity -csc(t)/sin(t) + cot(t)/tan(t) = -1 through a step-by-step process. We began by expressing all trigonometric functions in terms of sine and cosine, simplified the expression, combined fractions, applied the Pythagorean identity, and ultimately arrived at the right-hand side of the equation. This process demonstrates the importance of understanding fundamental trigonometric identities and mastering algebraic manipulation techniques.

Verifying trigonometric identities is not just an exercise in mathematical manipulation; it's a journey into the heart of trigonometry. By mastering this skill, you gain a deeper appreciation for the interconnectedness of trigonometric functions and their applications in various fields. The ability to simplify complex expressions and prove identities is invaluable in solving problems in calculus, physics, and engineering. Moreover, the logical reasoning and problem-solving skills honed through this process are transferable to other areas of mathematics and beyond.

Remember, practice is key to mastering trigonometric identity verification. Work through numerous examples, and don't be afraid to experiment with different approaches. The more you practice, the more comfortable you will become with recognizing patterns, applying identities, and simplifying expressions. With dedication and perseverance, you can conquer the world of trigonometric identities and unlock their power in solving a wide range of problems.

  • Trigonometric Identities
  • Verifying Identities
  • Cosecant (csc)
  • Sine (sin)
  • Cotangent (cot)
  • Tangent (tan)
  • Pythagorean Identity
  • Reciprocal Identities
  • Quotient Identities
  • Algebraic Manipulation
  • Simplifying Expressions
  • Mathematical Proof