Proving the Trigonometric Identity: sin2x n cot2x cos2x n tan2x n 1

In this article, we will delve into the intricate process of proving a specific trigonometric identity: sin2x n cot2x cos2x n tan2x n 1. This identity involves a combination of fundamental trigonometric functions and requires a methodical approach to verify its validity. This process is crucial for understanding more complex trigonometric relationships and will be presented in a clear, step-by-step manner.

Introduction to Trigonometric Functions

Before diving into the proof, it is essential to understand the definitions of the trigonometric functions involved:

sin2x sin2x cot2x (cos2x) / (sin2x) tan2x (sin2x) / (cos2x)

These definitions will be pivotal in simplifying the trigonometric expression.

Left Side Expansion

Let's begin by expanding the left side of the equation. The given identity is:

sin2x n cot2x cos2x n tan2x n 1

Step 1: Expand Each Term

sin2x n cot2x cos2x n tan2x

Using the definitions of cot2x and tan2x, we get:

sin2x n cot2x sin2x n (cos2x / sin2x)

cos2x n tan2x cos2x n (sin2x / cos2x)

Step 2: Simplify the Terms

Let's simplify the expressions step by step.

For the first term:

sin2x n (cos2x / sin2x)

Multiplying the second term by sin2x gives:

sin2x n cos2x / sin2x

Multiplying both terms by sin2x and then simplifying:

sin4x / sin2x n cos2x sin2x n cos2x

For the second term:

cos2x n (sin2x / cos2x)

Multiplying the second term by cos2x gives:

cos2x n sin2x / cos2x

Multiplying both terms by cos2x and then simplifying:

cos4x / cos2x n sin2x cos2x n sin2x

Combining the Results

Combining both simplified terms, we get:

sin2x n cos2x cos2x n sin2x

Further simplification yields:

(sin2x cos2x) n (cos2x sin2x)

Using the Pythagorean identity sin2x cos2x 1, we have:

1 n (1) 1 n

Right Side Verification

The right side of the equation is simply n 1. Thus, we can see that both sides of the equation are equal.

Final Result

Therefore, we conclude that the identity holds true:

sin2x n cot2x cos2x n tan2x n 1

Conclusion

Understanding and proving such trigonometric identities is fundamental for advanced mathematics and engineering applications. The step-by-step approach in this article provides a clear explanation of the proof.