Trigonometric Identity Proofs Cos²A + 1/(1 + Cot²A) = 1 And Csc⁴θ - Csc²θ = Cot⁴θ + Cot²θ

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Trigonometric identities are fundamental equations in trigonometry that hold true for all values of the variables involved. These identities are essential tools for simplifying trigonometric expressions, solving trigonometric equations, and understanding the relationships between different trigonometric functions. In this article, we will delve into the proofs of two important trigonometric identities: (i) cos²A + 1/(1 + cot²A) = 1 and (iii) csc⁴θ - csc²θ = cot⁴θ + cot²θ. Understanding these proofs not only reinforces your knowledge of trigonometric identities but also enhances your problem-solving skills in trigonometry.

Proof of Identity (i): cos²A + 1/(1 + cot²A) = 1

To prove the first identity, cos²A + 1/(1 + cot²A) = 1, we will start with the left-hand side (LHS) of the equation and manipulate it algebraically until it equals the right-hand side (RHS), which is 1. This approach involves utilizing fundamental trigonometric identities and algebraic manipulations to transform the expression. The main goal here is to show, step by step, how the left side of the equation can be simplified to match the right side, thus validating the identity.

Step-by-Step Proof

  1. Begin with the Left-Hand Side (LHS):

    • LHS = cos²A + 1/(1 + cot²A)

    We start by writing down the left-hand side of the equation. This is our starting point, and we will work on this expression to make it look like the right-hand side, which is simply 1. The key here is to identify the trigonometric terms and how they relate to each other, especially focusing on the cotangent function in this case.

  2. Use the Identity 1 + cot²A = csc²A:

    • We know the trigonometric identity 1 + cot²A = csc²A. Substitute this into the denominator:

      • LHS = cos²A + 1/csc²A

    This step is crucial. We are using a well-known Pythagorean identity that connects cotangent and cosecant. By substituting 1 + cot²A with csc²A, we simplify the expression and bring it closer to a form that we can further manipulate using other trigonometric relationships.

  3. Use the Identity 1/csc²A = sin²A:

    • Recall that csc A is the reciprocal of sin A, so 1/csc²A = sin²A. Substitute this into the equation:

      • LHS = cos²A + sin²A

    Here, we use the reciprocal identity of cosecant, which is sine. Replacing 1/csc²A with sin²A is a significant step because it transforms the expression into a sum of squares of cosine and sine, which we know has a direct relationship based on another fundamental Pythagorean identity.

  4. Apply the Pythagorean Identity cos²A + sin²A = 1:

    • We know the fundamental trigonometric identity cos²A + sin²A = 1. Therefore:

      • LHS = 1

    This is the final step in the proof. We use the most fundamental Pythagorean identity, cos²A + sin²A = 1. This identity is the cornerstone of many trigonometric proofs and simplifications. By applying this, we directly show that the left-hand side simplifies to 1.

  5. Conclusion:

    • Since LHS = 1 and RHS = 1, we have proven that cos²A + 1/(1 + cot²A) = 1.

    We conclude the proof by stating that since we have successfully manipulated the left-hand side of the equation to equal the right-hand side, we have proven the identity. This step provides a clear and concise statement of the result.

Importance of This Identity

This identity is a powerful tool in trigonometry, allowing for the simplification of expressions involving trigonometric functions. It highlights the interconnectedness of trigonometric functions and demonstrates how Pythagorean identities can be used to bridge these relationships. The identity is particularly useful in calculus and other advanced mathematical fields where trigonometric expressions need to be simplified for integration or differentiation.

Proof of Identity (iii): csc⁴θ - csc²θ = cot⁴θ + cot²θ

Now, let's move on to the second identity: csc⁴θ - csc²θ = cot⁴θ + cot²θ. This identity involves higher powers of trigonometric functions, which makes it a bit more complex but equally fascinating to prove. Similar to the previous proof, we will start with the left-hand side (LHS) and manipulate it to match the right-hand side (RHS). This involves strategic use of trigonometric identities and algebraic techniques. The aim is to demonstrate how the interplay between cosecant and cotangent functions, through known identities, leads to the equality stated in the identity.

Step-by-Step Proof

  1. Begin with the Left-Hand Side (LHS):

    • LHS = csc⁴θ - csc²θ

    We start by writing down the left-hand side of the equation. This is where our manipulation begins. The expression involves powers of the cosecant function, and our goal is to transform it into an expression involving cotangent functions.

  2. Factor out csc²θ:

    • Factor out csc²θ from the expression:

      • LHS = csc²θ(csc²θ - 1)

    This is a crucial step where we factor out the common term csc²θ. Factoring simplifies the expression and allows us to see a familiar term (csc²θ - 1) which we can relate to other trigonometric identities.

  3. Use the Identity 1 + cot²θ = csc²θ:

    • Rearrange the identity to get csc²θ - 1 = cot²θ. Substitute this into the equation:

      • LHS = csc²θ(cot²θ)

    Here, we use the Pythagorean identity 1 + cot²θ = csc²θ. By rearranging this identity, we get csc²θ - 1 = cot²θ, which allows us to replace the term in the parenthesis with cot²θ. This substitution is key to transforming the expression from cosecant to cotangent.

  4. Substitute csc²θ with 1 + cot²θ:

    • Use the identity csc²θ = 1 + cot²θ to substitute csc²θ:

      • LHS = (1 + cot²θ)(cot²θ)

    Now, we substitute csc²θ back with 1 + cot²θ. This step is important because it converts the entire expression into terms of cotangent, which is what we need to match the right-hand side of the identity.

  5. Distribute cot²θ:

    • Distribute cot²θ across the parentheses:

      • LHS = cot²θ + cot⁴θ

    This is a straightforward algebraic step where we distribute cot²θ across the terms inside the parenthesis. This step simplifies the expression and brings it closer to the form of the right-hand side.

  6. Rearrange the terms:

    • Rearrange the terms to match the RHS:

      • LHS = cot⁴θ + cot²θ

    We simply rearrange the terms to match the order on the right-hand side of the equation. This step is more for visual confirmation that we have indeed arrived at the right-hand side.

  7. Conclusion:

    • Since LHS = cot⁴θ + cot²θ and RHS = cot⁴θ + cot²θ, we have proven that csc⁴θ - csc²θ = cot⁴θ + cot²θ.

    We conclude the proof by stating that since we have successfully manipulated the left-hand side of the equation to equal the right-hand side, we have proven the identity. This step provides a clear and concise statement of the result.

Applications of This Identity

This identity is particularly useful in simplifying complex trigonometric expressions in calculus and physics problems. It demonstrates how factoring and applying Pythagorean identities can significantly reduce the complexity of an expression. The ability to transform expressions involving cosecants into expressions involving cotangents (and vice versa) is a valuable skill in advanced mathematical problem-solving. This identity also helps in understanding the symmetry and relationships between different trigonometric functions, which is crucial for more advanced topics in mathematics and physics.

Conclusion

In conclusion, we have successfully proven two important trigonometric identities: (i) cos²A + 1/(1 + cot²A) = 1 and (iii) csc⁴θ - csc²θ = cot⁴θ + cot²θ. These proofs highlight the power and elegance of trigonometric identities in simplifying expressions and solving problems. By understanding these identities and their proofs, you can enhance your problem-solving skills and gain a deeper appreciation for the interconnectedness of trigonometric functions. Trigonometric identities are not just theoretical constructs; they are essential tools for anyone working with mathematics, physics, and engineering. They provide a foundation for solving real-world problems involving angles, oscillations, and waves. Mastering these identities is a crucial step in any mathematical journey, enabling a more profound understanding of the mathematical world around us.

Through these detailed, step-by-step proofs, we've not only validated these identities but also illuminated the underlying principles of trigonometric manipulation. The ability to transform and simplify trigonometric expressions is a crucial skill in various fields, ranging from engineering to computer graphics. These proofs serve as a testament to the beauty and utility of trigonometric identities in mathematics and its applications. By understanding these identities, you unlock new avenues for problem-solving and gain a deeper appreciation for the elegance of mathematics.