Given:
A = sin^2(theta) * cos^4(theta)

To find:
The range of values for A for all values of theta.

Solution:

1. Simplifying the expression:
Using the trigonometric identity sin^2(theta) = 1 - cos^2(theta), we can rewrite the expression for A as:

A = (1 - cos^2(theta)) * cos^4(theta)
= cos^4(theta) - cos^6(theta)

2. Analyzing the range of values for A:
To determine the range of values for A, we need to consider the range of values for cos^4(theta) - cos^6(theta).

2.1 Lower bound for A:
To find the lower bound for A, we need to find the minimum value of cos^4(theta) - cos^6(theta). Since both cos^4(theta) and cos^6(theta) are non-negative, their difference will also be non-negative. Therefore, the minimum value of A is 0.

2.2 Upper bound for A:
To find the upper bound for A, we need to analyze the maximum value of cos^4(theta) - cos^6(theta). For any value of theta, cos^4(theta) will always be less than or equal to 1, and cos^6(theta) will always be less than or equal to cos^4(theta). Therefore, the maximum value of A occurs when cos^4(theta) = 1 and cos^6(theta) = cos^4(theta). In this case, A becomes:

A = 1 - cos^4(theta)

Since 0 ≤ cos^4(theta) ≤ 1, the maximum value of A is 1.

3. Conclusion:
From the analysis above, we can conclude that for all values of theta, the value of A lies between 0 and 1. Therefore, the correct option is:

3) 0 ≤ A ≤ 1

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