The Minimum Value of Sin2x · Cos2x · Tan2x · Sec2x · Cosec2x · Cot2x

Introduction

Understanding the minimum value of the expression sin2x · cos2x · tan2x · sec2x · cosec2x · cot2x is a fascinating exploration into the realms of trigonometry and inequalities. This expression involves multiple trigonometric functions and provides a rich ground to apply various mathematical techniques.

Step-by-Step Solution

Let's break down the given expression into its components and utilize trigonometric identities to simplify it. The expression is:

sin2x · cos2x · tan2x · sec2x · cosec2x · cot2x

Simplification Using Identities

First, let's rewrite the expression using the following trigonometric identities:

Applying these identities, the expression simplifies to:

1 · tan2x · sec2x · cosec2x · cot2x

Further Simplification

Next, we simplify further:

1 · tan2x · (1 tan2x) · (1 cot2x) · cot2x

Let s sin2x and c cos2x. Then, s c 1 and the expression becomes:

2 · (1/c c/1) - 1

Which simplifies to:

2 · (1/s 1/c) - 1

Since s · c 1, the expression further simplifies to:

2 (s c) - 1 2 - 1 7

Thus, the minimum value of the expression is 7.

Application of AM-GM Inequality

Although the AM-GM (Arithmetic Mean-Geometric Mean) inequality could provide a lower bound, it's important to note that this method doesn't yield the exact minimum in this case. The AM-GM inequality requires all terms to be non-negative, and in this scenario, the terms sec2x - 1 and cosec2x - 1 can be negative. Let's rewrite the expression:

3 · (2 tan2x 2 cot2x)

Using the AM-GM property, we find:

2 tan2x 2 cot2x ≥ 2

Thus, the minimum value is:

3 · 2 6

However, the actual minimum obtained using simplification is 7, with the expression achieving its minimum when:

sin^2(2x) 1

This occurs at x π/4, 3π/4.

Conclusion

The minimum value of the complex trigonometric expression sin2x · cos2x · tan2x · sec2x · cosec2x · cot2x is 7.