Prove that : sec⁴ theta(1-sin⁴theta)-2 tan²  theta = 1

To prove the identity:

\[
\sec^4 \theta (1 - \sin^4 \theta) - 2 \tan^2 \theta = 1
\]

### Step 1: Expand \(1 - \sin^4 \theta\)

We can express \(1 - \sin^4 \theta\) as a difference of squares:

\[
1 - \sin^4 \theta = (1 - \sin^2 \theta)(1 + \sin^2 \theta)
\]

We know that \(1 - \sin^2 \theta = \cos^2 \theta\). Therefore:

\[
1 - \sin^4 \theta = \cos^2 \theta(1 + \sin^2 \theta)
\]

### Step 2: Substitute into the original expression

Substitute \(1 - \sin^4 \theta = \cos^2 \theta(1 + \sin^2 \theta)\) into the original equation:

\[
\sec^4 \theta \cdot \cos^2 \theta(1 + \sin^2 \theta) - 2 \tan^2 \theta = 1
\]

### Step 3: Simplify the expression

Recall that \(\sec \theta = \frac{1}{\cos \theta}\), so:

\[
\sec^4 \theta \cdot \cos^2 \theta = \frac{1}{\cos^4 \theta} \cdot \cos^2 \theta = \frac{1}{\cos^2 \theta} = \sec^2 \theta
\]

Thus, the expression becomes:

\[
\sec^2 \theta(1 + \sin^2 \theta) - 2 \tan^2 \theta = 1
\]

### Step 4: Expand and simplify

Expand the left-hand side:

\[
\sec^2 \theta + \sec^2 \theta \cdot \sin^2 \theta - 2 \tan^2 \theta
\]

We know that \(\tan^2 \theta = \sec^2 \theta - 1\), so substitute that in:

\[
\sec^2 \theta + \sec^2 \theta \cdot \sin^2 \theta - 2(\sec^2 \theta - 1)
\]

Simplify the expression:

\[
\sec^2 \theta + \sec^2 \theta \cdot \sin^2 \theta - 2\sec^2 \theta + 2
\]

Combine like terms:

\[
\sec^2 \theta \cdot \sin^2 \theta - \sec^2 \theta + 2
\]

Factor out \(\sec^2 \theta\):

\[
\sec^2 \theta (\sin^2 \theta - 1) + 2
\]

Since \(\sin^2 \theta - 1 = -\cos^2 \theta\), this simplifies to:

\[
-\sec^2 \theta \cdot \cos^2 \theta + 2
\]

Finally:

\[
-1 + 2 = 1
\]

This confirms that the identity is correct:

\[
\sec^4 \theta (1 - \sin^4 \theta) - 2 \tan^2 \theta = 1
\]

The identity is proven!

Would you like more details or have any questions? Here are some related problems:

1. Prove the identity \(\sec^2 \theta - \tan^2 \theta = 1\).
2. Simplify \(\sec^4 \theta - \tan^4 \theta\).
3. Show that \(\sin^2 \theta + \cos^2 \theta = 1\).
4. Prove the identity \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\).
5. Simplify \(\sec^4 \theta - 2\sec^2 \theta + 1\).

**Tip:** In trigonometric identities, the Pythagorean identities (\(\sin^2 \theta + \cos^2 \theta = 1\), \(\sec^2 \theta = 1 + \tan^2 \theta\)) are very useful for simplification.

Learn how to prove the trigonometric identity sec^4 theta(1-sin^4 theta)-2 tan^2 theta = 1 step-by-step. This problem involves applying trigonometric identities and theorems, including the secant and tangent identities. Suitable for high school students.

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