A Mathematical Proof: How tan 40°tan 41° … tan 49°tan 50° Equals 1

Trigonometry is a rich field of mathematics with countless intriguing identities and proofs. One such fascinating identity involves the tangent function, specifically the product of tangents from specific angles. This article explores the proof of the equation tan 40°tan 41°tan 49°tan 50° 1. The proof relies on the application of trigonometric identities and understanding the behavior of the tangent and cotangent functions.

Introduction to Trigonometric Identities

Before diving into the proof, it is important to understand some basic trigonometric identities and properties. The tangent function, denoted as tanθ, is defined as the ratio of the sine and cosine functions: tanθ sinθ / cosθ. The cotangent function, denoted as cotθ, is the reciprocal of the tangent function: cotθ 1 / tanθ. Additionally, the cotangent function can also be expressed in terms of sine and cosine: cotθ cosθ / sinθ.

Step-by-Step Proof

Now, let's proceed with the step-by-step proof of the equation:

1. Express the product using cotangent:

The product of tangents can be transformed using the cotangent function. Recall that cotangent and tangent are reciprocals of each other, so:

tan 40°tan 41°tan 49°tan 50° tan 40°tan 41°tan 90° - 41tan 90° - 40tan 90° - 40tan 90° - 41

Using the identity tan(90° - θ) cotθ, we rewrite the terms:

tan 40°tan 41°tan 90° - 41tan 90° - 40tan 90° - 40tan 90° - 41 tan 40°cot 41°tan 49°cot 40°

2. Simplify the product:

Next, we simplify the product by recognizing that tanθcotθ 1. Hence:

tan 40°cot 41°tan 49°cot 40° (tan 40°cot 40°)(tan 41°cot 41°)tan 49°

3. Apply the identity:

Knowing that tanθcotθ 1, we can replace each pair of tangent and cotangent with 1:

(tan 40°cot 40°)(tan 41°cot 41°)tan 49° 11tan 49°

4. Final simplification:

Since tan 45° 1, we can conclude that the product of the tangent terms is 1:

11tan 49° 111 1

This completes the proof that tan 40°tan 41°tan 49°tan 50° 1.

Conclusion

The proof of the equation tan 40°tan 41°tan 49°tan 50° 1 involves the manipulation of trigonometric identities, particularly the relationships between tangent and cotangent. Understanding these identities and their applications can provide valuable insights into the behavior of trigonometric functions and enhance one's problem-solving skills in mathematics. Applying such proofs can also be beneficial in various fields such as engineering, physics, and even computer science where trigonometric functions play a significant role.

Key Takeaways:

Trigonometric Identities: The proof heavily relies on the identities tan(90° - θ) cotθ and tanθcotθ 1. Behavior of Tangent and Cotangent: Understanding the reciprocal relationship between tangent and cotangent is crucial. Application of Identities: Effective use of mathematical identities can simplify complex trigonometric expressions.