Proof:

To prove the given statement, we will start by simplifying the expression sin^4(theta) / a * cos^4(theta) / b = 1 / (a * b).

Step 1: Simplifying the expression

Let's start by simplifying the left-hand side of the equation:

sin^4(theta) / a * cos^4(theta) / b

Using the property (a * b) / c = a * (b / c), we can rewrite the expression as:

(sin^4(theta) * cos^4(theta)) / (a * b)

Step 2: Applying trigonometric identity

We can apply the trigonometric identity sin^2(theta) + cos^2(theta) = 1 to simplify the expression further.

(sin^2(theta) * cos^2(theta))^2 / (a * b)

(sin^2(theta))^2 * (cos^2(theta))^2 / (a * b)

(sin^4(theta) * cos^4(theta)) / (a * b)

We can see that the expression has simplified back to the original expression.

Step 3: Rearranging the expression

Now, let's rearrange the expression sin^8(theta) / a^3 * cos^8(theta) / b^3 = 1 / (a * b)^3.

We can rewrite the left-hand side as:

(sin^4(theta))^2 / a^3 * (cos^4(theta))^2 / b^3

(sin^4(theta) * cos^4(theta))^2 / (a^3 * b^3)

Now, we can substitute the simplified expression from step 2:

1^2 / (a^3 * b^3)

1 / (a^3 * b^3)

Step 4: Conclusion

We have successfully shown that sin^8(theta) / a^3 * cos^8(theta) / b^3 = 1 / (a * b)^3.

Therefore, the given statement is proved.