Proof:

To prove that sec^8a - 1 / sec^4a - 1 = tan^8a / tan^2a, we can use the trigonometric identities and simplify both sides of the equation.

Using Trigonometric Identities:

1. We know that sec^2a - 1 = tan^2a. Rearranging this equation, we get sec^2a = tan^2a + 1.

2. Squaring both sides of the equation, we obtain sec^4a = (tan^2a + 1)^2.

3. Expanding the equation, we have sec^4a = tan^4a + 2tan^2a + 1.

4. Multiplying both sides of the equation by sec^4a, we get sec^8a = tan^8a + 2tan^6a + tan^4a.

Simplifying the Equation:

Now, let's substitute the values of sec^8a and sec^4a into the given equation and simplify it step by step.

sec^8a - 1 / sec^4a - 1 = tan^8a / tan^2a

Using the trigonometric identity sec^8a = tan^8a + 2tan^6a + tan^4a, we can rewrite the equation as:

(tan^8a + 2tan^6a + tan^4a - 1) / sec^4a - 1 = tan^8a / tan^2a

Next, let's simplify the numerator of the left-hand side of the equation:

(tan^8a + 2tan^6a + tan^4a - 1) = (tan^8a + tan^4a) + 2tan^6a - 1

= tan^4a(tan^4a + 1) + 2tan^6a - 1

Using the identity sec^4a = tan^4a + 2tan^2a + 1, we can substitute sec^4a in the denominator:

= tan^4a(tan^4a + 1) + 2tan^6a - 1 / (tan^4a + 2tan^2a + 1) - 1

Simplifying the denominator, we get:

= tan^4a(tan^4a + 1) + 2tan^6a - 1 / (tan^4a + 2tan^2a) - 1

Now, let's cancel out the common factors in the numerator and denominator:

= tan^4a + 2tan^6a - 1 / tan^4a + 2tan^2a

Finally, we can rewrite the equation in a simplified form:

= tan^8a / tan^2a

Hence, we have proved that sec^8a - 1 / sec^4a - 1 = tan^8a / tan^2a.