Introduction

In this article, we will delve into the detailed process of determining the minimum value of the expression sin^6 x cos^6 x. We will employ various trigonometric identities and algebraic manipulations to systematically derive the solution. Let's begin.

1. Setting up the Expression

Consider the expression sin^6 x cos^6 x. The challenge lies in finding its minimum value, which can be approached through a series of algebraic transformations and trigonometric identities.

1.1 Trigonometric Identity Review

Recall the trigonometric identity for the product of cubes:

a^3 b^3 a b (a^2 - a b b^2)

Let's set:

a sin^2 x and b cos^2 x

2. Expressing the Original Expression

Substituting a sin^2 x and b cos^2 x into our identity, we get:

sin^6 x cos^6 x sin^2 x (cos^2 x) (sin^2 x - sin^2 x cos^2 x cos^2 x)

Let's simplify further:

sin^6 x cos^6 x sin^2 x (cos^2 x) (1 - 3 sin^2 x cos^2 x)

3. Further Simplification

Now, let's use the double-angle identity to express sin^2 x cos^2 x in a more manageable form:

sin^2 x cos^2 x (1/4) sin^2 2x

Substitute this into our expression:

sin^6 x cos^6 x sin^2 x (1/4) sin^2 2x (1 - 3(1/4) sin^2 2x)

Simplifying:

sin^6 x cos^6 x (1/4) sin^2 x sin^2 2x (1 - 3/4 sin^2 2x)

4. Finding the Minimum Value

To find the minimum value, we need to analyze the range of the expression 1 - 3/4 sin^2 2x.

Note that 0 ≤ sin^2 2x ≤ 1, so 0 ≤ 3/4 sin^2 2x ≤ 3/4, and hence 1 - 3/4 ≤ 1 - 3/4 sin^2 2x ≤ 1.

The minimum value occurs when sin^2 2x 1, giving us:

1 - 3/4 1/4

5. Conclusion

The minimum value of the expression sin^6 x cos^6 x is 1/4.

[boxed{1/4}]