Sin square theta + Cos raise to 4 theta minus cos squared theta the whole divided by cos square theta 

Let's simplify the given expression:

\[
\frac{\sin^2\theta + \cos^4\theta - \cos^2\theta}{\cos^2\theta}
\]

### Step 1: Break down the numerator
Let's first consider the numerator:

\[
\sin^2\theta + \cos^4\theta - \cos^2\theta
\]

We can rearrange the terms as follows:

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

### Step 2: Factor the expression inside the parenthesis
The expression \(\cos^4\theta - \cos^2\theta\) can be factored as:

\[
\cos^2\theta(\cos^2\theta - 1)
\]

Thus, the numerator becomes:

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

### Step 3: Substitute \(\sin^2\theta\) using the identity
We know the identity:

\[
\sin^2\theta = 1 - \cos^2\theta
\]

So, substitute this into the numerator:

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

Simplifying further:

\[
1 - \cos^2\theta + \cos^4\theta - \cos^2\theta
\]

Combine like terms:

\[
1 - 2\cos^2\theta + \cos^4\theta
\]

### Step 4: Simplify the entire expression
Now, divide the entire numerator by \(\cos^2\theta\):

\[
\frac{1 - 2\cos^2\theta + \cos^4\theta}{\cos^2\theta}
\]

This can be rewritten as:

\[
\frac{1}{\cos^2\theta} - 2 + \cos^2\theta
\]

### Step 5: Simplify further
The term \(\frac{1}{\cos^2\theta}\) is the secant function squared:

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

So, the final simplified expression is:

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

This is the simplest form of the expression.

Would you like more details on any of the steps, or do you have any questions?

### Related Questions:
1. How does the identity \(\sin^2\theta + \cos^2\theta = 1\) help in trigonometric simplifications?
2. What are some common factoring techniques used in trigonometric expressions?
3. How is the secant function defined and related to the cosine function?
4. What are other trigonometric identities involving \(\cos^2\theta\) and \(\sin^2\theta\)?
5. Can this expression be further simplified using any other identities?

**Tip:** Always look for trigonometric identities that can simplify complex expressions, such as the Pythagorean identity.

Sin square theta + Cos raise to 4 theta minus cos squared theta the whole divided by cos square theta

Explore the step-by-step process of simplifying the trigonometric expression Sin^2(θ) + Cos^4(θ) - Cos^2(θ) divided by Cos^2(θ). This solution highlights key trigonometric identities and factorization techniques. Ideal for students in Grades 10-12.

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