 sin square theta / cos square theta minus 1 minus Cos raise to 4 theta divided by whole square theta

The expression you've provided can be written as:

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

Let's simplify this step by step:

### Step 1: Simplify \(\frac{\sin^2 \theta}{\cos^2 \theta}\)
\[
\frac{\sin^2 \theta}{\cos^2 \theta} = \tan^2 \theta
\]

So, the expression becomes:
\[
\tan^2 \theta - 1 - \frac{\cos^4 \theta}{\theta^2}
\]

### Step 2: Simplify \(\tan^2 \theta - 1\)
We know the identity:
\[
\tan^2 \theta - 1 = \sec^2 \theta - 2
\]

Thus, the expression now is:
\[
\sec^2 \theta - 2 - \frac{\cos^4 \theta}{\theta^2}
\]

### Final Expression:
\[
\sec^2 \theta - 2 - \frac{\cos^4 \theta}{\theta^2}
\]

This is the simplified form of the given expression.

Do you have any questions or would you like a further breakdown?

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Here are some related questions:

1. What are the key trigonometric identities?
2. How do you derive the identity \(\tan^2 \theta = \sec^2 \theta - 1\)?
3. How can \(\cos^4 \theta\) be expressed using double angle identities?
4. What is the relationship between \(\sec \theta\) and \(\cos \theta\)?
5. How can trigonometric identities simplify integrals or derivatives?

**Tip:** Knowing trigonometric identities can make solving complex expressions easier!

sin square theta / cos square theta minus 1 minus Cos raise to 4 theta divided by whole square theta

This page provides a detailed step-by-step solution to simplify the trigonometric expression sin^2(θ)/cos^2(θ) - 1 - cos^4(θ)/θ^2. It involves key trigonometric identities like tan^2(θ) = sec^2(θ) - 1. Suitable for students in Grades 10-12.

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