Sin square theta divided by cos square theta minus 1 minus Cos raise to 4 theta divided by cos square theta

To solve the expression \(\frac{\sin^2 \theta}{\cos^2 \theta} - 1 - \frac{\cos^4 \theta}{\cos^2 \theta}\), let's break it down step by step:

### Step 1: Simplify the first term
The first term is \(\frac{\sin^2 \theta}{\cos^2 \theta}\). This can be rewritten using the identity for tangent:

\[
\frac{\sin^2 \theta}{\cos^2 \theta} = \tan^2 \theta
\]

### Step 2: Simplify the second term
The second term is \(1\), which remains unchanged.

### Step 3: Simplify the third term
The third term is \(\frac{\cos^4 \theta}{\cos^2 \theta}\). This can be simplified by canceling out \(\cos^2 \theta\) in the numerator and the denominator:

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

### Step 4: Combine the simplified terms
Now, substitute the simplified expressions back into the original equation:

\[
\tan^2 \theta - 1 - \cos^2 \theta
\]

### Step 5: Apply a trigonometric identity
We can use the trigonometric identity \(\tan^2 \theta = \sec^2 \theta - 1\) to further simplify:

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

Simplify further:

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

### Final Expression
The simplified expression is:

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

Would you like further details on this solution, or do you have any questions?

### 5 Related Questions:
1. How is the identity \(\tan^2 \theta = \sec^2 \theta - 1\) derived?
2. What are other useful trigonometric identities for simplifying expressions?
3. How can we interpret \(\sec^2 \theta - \cos^2 \theta\) geometrically?
4. What are the steps to simplify more complex trigonometric expressions?
5. How does this trigonometric expression relate to others involving sine and cosine?

### Tip:
Always look for opportunities to apply trigonometric identities, as they can significantly simplify complex expressions.

Explore the step-by-step solution to simplifying the trigonometric expression sin^2(θ)/cos^2(θ) - 1 - cos^4(θ)/cos^2(θ). This problem involves key trigonometric identities and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation