3sec^2theta tan^2 theta + tan^6 theta-sec^6 theta

We can simplify the expression \( 3\sec^2\theta \tan^2\theta + \tan^6\theta - \sec^6\theta \) using trigonometric identities.

Let's break it down step by step:

### Expression:
\[
3\sec^2\theta \tan^2\theta + \tan^6\theta - \sec^6\theta
\]

### Step 1: Identify relevant identities
We know the following trigonometric identities:
1. \( \sec^2\theta = 1 + \tan^2\theta \)
2. \( \sec^6\theta = (1 + \tan^2\theta)^3 \)

### Step 2: Substitute \( \sec^2\theta \) in the expression
Substituting \( \sec^2\theta = 1 + \tan^2\theta \) into the expression:

\[
3(1 + \tan^2\theta) \tan^2\theta + \tan^6\theta - (1 + \tan^2\theta)^3
\]

### Step 3: Expand the terms
First, expand \( 3(1 + \tan^2\theta) \tan^2\theta \):
\[
3\tan^2\theta + 3\tan^4\theta
\]
Next, expand \( (1 + \tan^2\theta)^3 \) using the binomial theorem:
\[
(1 + \tan^2\theta)^3 = 1 + 3\tan^2\theta + 3\tan^4\theta + \tan^6\theta
\]

### Step 4: Substitute back and simplify
Substitute the expanded forms back into the expression:
\[
3\tan^2\theta + 3\tan^4\theta + \tan^6\theta - \left(1 + 3\tan^2\theta + 3\tan^4\theta + \tan^6\theta\right)
\]

Now, combine like terms:
\[
(3\tan^2\theta + 3\tan^4\theta + \tan^6\theta) - (1 + 3\tan^2\theta + 3\tan^4\theta + \tan^6\theta)
\]

This simplifies to:
\[
0 - 1 = -1
\]

### Final Answer:
\[
3\sec^2\theta \tan^2\theta + \tan^6\theta - \sec^6\theta = -1
\]

The simplified expression is \(-1\).

Would you like to explore the steps in more detail or have any other questions?

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**Related Questions:**
1. How does the identity \( \sec^2\theta = 1 + \tan^2\theta \) help in simplifying trigonometric expressions?
2. What is the binomial theorem, and how does it apply to \( (1 + \tan^2\theta)^3 \)?
3. How do you expand trigonometric expressions using binomial expansions?
4. Can you provide more examples where the identity \( \sec^2\theta = 1 + \tan^2\theta \) is used?
5. What other trigonometric identities are important for simplifying expressions?
6. How would the expression change if \( \theta \) had a specific value?
7. How can we check if our simplification of the expression is correct?
8. What is the general approach to simplifying complex trigonometric expressions?

**Tip:** When simplifying trigonometric expressions, always look for identities that can replace complex terms with simpler ones.

Learn how to simplify the trigonometric expression 3sec^2(theta) tan^2(theta) + tan^6(theta) - sec^6(theta) step-by-step. This problem utilizes trigonometric identities like secant and tangent, along with the binomial theorem for expansion. Suitable for high school students studying trigonometry.

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