How To Prove The Cofunction Identity Tan(π/2 - U) = Cot(u) Using The Addition And Subtraction Formulas For Sine And Cosine? Since Tan(π/2) Is Undefined, How To Use A Reciprocal Identity To Help?

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In this article, we will delve into the fascinating world of trigonometric identities and explore a proof for the cofunction identity: tan(π/2 - u) = cot(u). This identity reveals an elegant relationship between the tangent and cotangent functions, demonstrating how they are intertwined through a simple transformation involving π/2. To unravel this identity, we will employ the powerful tools of the Addition and Subtraction Formulas for sine and cosine, along with a touch of reciprocal identity magic. Let's embark on this mathematical journey!

Understanding Cofunction Identities

Before diving into the proof, let's first grasp the essence of cofunction identities. In trigonometry, cofunctions are pairs of functions that exhibit a special relationship: the value of one function at an angle is equal to the value of its cofunction at the complementary angle. Complementary angles are those that add up to π/2 (90 degrees). The primary cofunction pairs are sine and cosine, tangent and cotangent, and secant and cosecant.

The cofunction identity we aim to prove, tan(π/2 - u) = cot(u), exemplifies this relationship between tangent and cotangent. It states that the tangent of the angle (π/2 - u) is equivalent to the cotangent of the angle u. This identity has profound implications in simplifying trigonometric expressions and solving trigonometric equations.

Leveraging Addition and Subtraction Formulas

Our proof hinges on the fundamental Addition and Subtraction Formulas for sine and cosine. These formulas provide a way to express trigonometric functions of sums and differences of angles in terms of trigonometric functions of the individual angles. Let's recall these formulas:

  • Sine Addition Formula: sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
  • Sine Subtraction Formula: sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
  • Cosine Addition Formula: cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
  • Cosine Subtraction Formula: cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

These formulas are indispensable tools in trigonometric manipulations, and we will see how they pave the way for proving our cofunction identity.

The Proof: A Step-by-Step Unveiling

Now, let's embark on the core of our endeavor: proving the identity tan(π/2 - u) = cot(u). We'll meticulously dissect the problem, applying the aforementioned formulas and identities to reach our desired conclusion.

  1. Expressing Tangent in terms of Sine and Cosine: Our starting point is the fundamental definition of the tangent function: tan(x) = sin(x) / cos(x). Applying this to our identity, we get:

    tan(π/2 - u) = sin(π/2 - u) / cos(π/2 - u)

    This step transforms our problem into one involving sine and cosine, setting the stage for the application of the Addition and Subtraction Formulas.

  2. Applying the Sine and Cosine Subtraction Formulas: We now invoke the Sine and Cosine Subtraction Formulas to expand sin(π/2 - u) and cos(π/2 - u):

    sin(π/2 - u) = sin(π/2)cos(u) - cos(π/2)sin(u)

    cos(π/2 - u) = cos(π/2)cos(u) + sin(π/2)sin(u)

    These formulas allow us to express the sine and cosine of the difference of angles in terms of the sines and cosines of the individual angles.

  3. Evaluating Trigonometric Functions at π/2: Next, we utilize our knowledge of the values of sine and cosine at π/2:

    sin(π/2) = 1

    cos(π/2) = 0

    Substituting these values into our expanded expressions, we obtain:

    sin(π/2 - u) = (1)cos(u) - (0)sin(u) = cos(u)

    cos(π/2 - u) = (0)cos(u) + (1)sin(u) = sin(u)

    This simplification unveils a crucial relationship: the sine of (π/2 - u) is equal to the cosine of u, and the cosine of (π/2 - u) is equal to the sine of u.

  4. Substituting Back into the Tangent Expression: Now, we substitute these simplified expressions back into our equation for tan(π/2 - u):

    tan(π/2 - u) = cos(u) / sin(u)

    This step brings us closer to our desired identity.

  5. Applying the Reciprocal Identity: Finally, we employ the reciprocal identity for cotangent: cot(x) = cos(x) / sin(x). Applying this to our expression, we get:

    tan(π/2 - u) = cot(u)

    This elegant final step completes our proof, demonstrating that the tangent of (π/2 - u) is indeed equal to the cotangent of u.

Conclusion: The Beauty of Trigonometric Identities

In this exploration, we have successfully proven the cofunction identity tan(π/2 - u) = cot(u) using the Addition and Subtraction Formulas for sine and cosine. This journey highlights the interconnectedness of trigonometric functions and the power of trigonometric identities in simplifying expressions and revealing hidden relationships. The cofunction identity we've proven showcases the elegant symmetry inherent in trigonometry, where functions and their cofunctions dance in harmony across the unit circle. Mastering these identities is crucial for success in trigonometry and related fields, as they provide a powerful toolkit for solving problems and deepening our understanding of the mathematical world.

This exploration of the cofunction identity serves as a testament to the beauty and power of trigonometric relationships. By understanding these identities and mastering the techniques used to prove them, we equip ourselves with essential tools for navigating the world of mathematics and beyond. The journey through trigonometric proofs not only sharpens our problem-solving skills but also deepens our appreciation for the elegance and interconnectedness of mathematical concepts.