This trigonometric identity can be proved using the Pythagorean identity for sine and cosine, which states that sin^2x + cos^2x = 1.

Starting with the given equation:

sin^4a + sin^2a * cos^2a = 1 - cos^2a

We can manipulate the equation using the Pythagorean identity by substituting sin^2a with 1 - cos^2a:

(sin^2a)^2 + sin^2a * cos^2a = 1 - cos^2a

Now, let's simplify the left side by expanding (sin^2a)^2:

(sin^2a)^2 = sin^4a

Substitute sin^2a with 1 - cos^2a:

sin^4a + (1 - cos^2a) * cos^2a = 1 - cos^2a

Expand the product on the left side:

sin^4a + cos^2a - cos^4a = 1 - cos^2a

Rearrange the terms to get a more familiar form:

sin^4a - cos^4a = 1 - 2cos^2a

Now, apply the difference of squares identity:

(sin^2a + cos^2a)(sin^2a - cos^2a) = 1 - 2cos^2a

Remembering the Pythagorean identity (sin^2x + cos^2x = 1), the term in brackets simplifies to 1:

1 * (sin^2a - cos^2a) = 1 - 2cos^2a

This simplifies to:

sin^2a - cos^2a = 1 - 2cos^2a

Therefore, the given trigonometric identity:

sin^4a + sin^2a * cos^2a = 1 - cos^2a

Has been correctly proven.