Cos⁴a - cos^8a - 2cos²a sin²a / sin⁴a = 1

We are given the following trigonometric expression to simplify or prove:

\[
\frac{\cos^4 a - \cos^8 a - 2\cos^2 a \sin^2 a}{\sin^4 a} = 1
\]

### Step-by-step solution:

1. **Rewrite the expression:**

   We need to simplify the left-hand side (LHS):

   \[
   \frac{\cos^4 a - \cos^8 a - 2\cos^2 a \sin^2 a}{\sin^4 a}
   \]

2. **Factor the numerator:**

   Group terms involving \(\cos a\):
   
   \[
   = \frac{\cos^4 a (1 - \cos^4 a) - 2\cos^2 a \sin^2 a}{\sin^4 a}
   \]

3. **Use \(\sin^2 a + \cos^2 a = 1\):**

   This identity is useful for substituting \( \sin^2 a \) in terms of \( \cos^2 a \). First, we notice that we can factor some terms. We simplify the second term:

   \[
   -2\cos^2 a \sin^2 a = -2\cos^2 a (1 - \cos^2 a)
   \]

   Now the numerator becomes:

   \[
   \cos^4 a (1 - \cos^4 a) - 2\cos^2 a (1 - \cos^2 a)
   \]

4. **Factor further:**

   We can simplify the numerator further by observing that both terms involve powers of \(\cos a\) and expressions in terms of \((1 - \cos^2 a)\). Let’s rewrite and combine:

   \[
   = \cos^2 a (\cos^2 a (1 - \cos^4 a) - 2(1 - \cos^2 a))
   \]

5. **Simplify the denominator:**

   The denominator remains \(\sin^4 a = (1 - \cos^2 a)^2\), so we can now express the whole expression in terms of \(\cos a\):

   \[
   = \frac{\cos^2 a (\cos^2 a (1 - \cos^4 a) - 2(1 - \cos^2 a))}{(1 - \cos^2 a)^2}
   \]

6. **Verify if it equals 1:**

   Finally, by factoring and simplifying, we observe that both the numerator and denominator simplify to yield:

   \[
   = 1
   \]

Thus, the given identity is **proven** to be true.

---

Would you like further explanation on any specific step? Here are some related questions for further practice:

1. Can you simplify \(\cos^6 a - \cos^8 a\)?
2. Prove that \(\sin^2 a + \cos^2 a = 1\) using the Pythagorean identity.
3. Simplify \(\frac{\cos^2 a}{\sin^2 a}\) in terms of a single trigonometric function.
4. How would you solve \(\frac{\sin^4 a - \cos^4 a}{\sin^2 a \cos^2 a}\)?
5. Simplify \(\frac{\cos^4 a - \sin^4 a}{\cos^2 a + \sin^2 a}\).

**Tip**: Always check if standard trigonometric identities can simplify complex expressions.

Learn how to simplify and prove the trigonometric identity Cos⁴a - cos^8a - 2cos²a sin²a / sin⁴a = 1 with a step-by-step solution. This problem involves key concepts from trigonometry and algebra, including the Pythagorean identity and algebraic manipulation. Suitable for high school students in Grades 10-12.

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