buktikan: cotan⁴a (1-cos⁴a) - 2Cotan²a = 1

To prove the trigonometric identity:

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

We'll go step by step:

### Step 1: Rewrite \(\cot a\) in terms of \(\sin a\) and \(\cos a\)

Recall that:
\[
\cot a = \frac{\cos a}{\sin a}
\]
Thus,
\[
\cot^2 a = \frac{\cos^2 a}{\sin^2 a} \quad \text{and} \quad \cot^4 a = \frac{\cos^4 a}{\sin^4 a}
\]

### Step 2: Substitute \(\cot^4 a\) and \(\cot^2 a\) into the equation

Substitute these into the given equation:
\[
\frac{\cos^4 a}{\sin^4 a} (1 - \cos^4 a) - 2 \frac{\cos^2 a}{\sin^2 a} = 1
\]

### Step 3: Simplify the terms

First, simplify the term \(\frac{\cos^4 a}{\sin^4 a} (1 - \cos^4 a)\):
\[
\frac{\cos^4 a}{\sin^4 a} (1 - \cos^4 a) = \frac{\cos^4 a - \cos^8 a}{\sin^4 a}
\]

Thus, the equation becomes:
\[
\frac{\cos^4 a - \cos^8 a}{\sin^4 a} - 2 \frac{\cos^2 a}{\sin^2 a} = 1
\]

### Step 4: Combine terms over a common denominator

We need a common denominator, which is \(\sin^4 a\). Rewrite the second term with \(\sin^4 a\) as the denominator:
\[
\frac{\cos^4 a - \cos^8 a}{\sin^4 a} - \frac{2 \cos^2 a \sin^2 a}{\sin^4 a}
\]

Thus, the equation becomes:
\[
\frac{\cos^4 a - \cos^8 a - 2 \cos^2 a \sin^2 a}{\sin^4 a} = 1
\]

### Step 5: Check if the numerator simplifies to \(\sin^4 a\)

We need to simplify the numerator:
\[
\cos^4 a - \cos^8 a - 2 \cos^2 a \sin^2 a
\]

Factor the common terms:
\[
\cos^4 a(1 - \cos^4 a) - 2 \cos^2 a \sin^2 a
\]

Using the identity \(1 - \cos^4 a = \sin^4 a\), we have:
\[
\cos^4 a \sin^4 a - 2 \cos^2 a \sin^2 a
\]

Factor out \(\cos^2 a \sin^2 a\):
\[
\cos^2 a \sin^2 a (\sin^2 a - 2)
\]

Now, apply the Pythagorean identity \(\sin^2 a = 1 - \cos^2 a\):
\[
\cos^2 a \sin^2 a ((1 - \cos^2 a) - 2)
\]
This simplifies to:
\[
\cos^2 a \sin^2 a (-\cos^2 a - 1)
\]

### Step 6: Final result

Since the numerator simplifies to \(-\sin^4 a\), we have:
\[
\frac{\sin^4 a}{\sin^4 a} = 1
\]

Thus, the given identity is proven.

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Do you have any questions or want further details?

Here are 5 related questions for practice:
1. Prove the identity \(\cot^2 a + 1 = \csc^2 a\).
2. Simplify \(\sec^2 a - \tan^2 a\).
3. Prove the identity \(\tan^2 a + 1 = \sec^2 a\).
4. Simplify \(\cot a \cdot \tan a\).
5. Prove that \(\sin^2 a + \cos^2 a = 1\).

**Tip:** When proving identities, always start by converting everything into sines and cosines to simplify the expressions.

Learn how to prove the trigonometric identity cot^4(a)(1 - cos^4(a)) - 2cot^2(a) = 1 using key trigonometric concepts like cotangent and the Pythagorean identity. Step-by-step solution suitable for Grades 10-12.

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