To verify the identity \(\cos^2 \theta (1 + \tan^2 \theta) = 1\), we can use trigonometric identities. Here’s the step-by-step process:

1. Recall the identity for \(\tan \theta\):
   \[
   \tan \theta = \frac{\sin \theta}{\cos \theta}
   \]
   Therefore,
   \[
   \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}
   \]

2. Substitute \(\tan^2 \theta\) into the given equation:
   \[
   \cos^2 \theta \left(1 + \frac{\sin^2 \theta}{\cos^2 \theta}\right) = 1
   \]

3. Combine the terms inside the parentheses:
   \[
   1 + \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta}
   \]

4. Recall the Pythagorean identity:
   \[
   \sin^2 \theta + \cos^2 \theta = 1
   \]

5. Substitute the Pythagorean identity into the equation:
   \[
   \cos^2 \theta \left(\frac{1}{\cos^2 \theta}\right) = 1
   \]

6. Simplify the expression:
   \[
   \cos^2 \theta \cdot \frac{1}{\cos^2 \theta} = 1
   \]

7. This simplifies to:
   \[
   1 = 1
   \]

This confirms that the given identity is true.

Would you like more details or have any questions?

Here are some related questions:

1. How can you derive the Pythagorean identities from the unit circle?
2. What are some common trigonometric identities involving tangent?
3. How do you prove trigonometric identities using algebraic manipulation?
4. What is the relationship between secant and cosine functions?
5. How do you simplify expressions involving multiple trigonometric identities?
6. Can you explain why \(\cos^2 \theta + \sin^2 \theta = 1\) is fundamental in trigonometry?
7. How do trigonometric identities help in solving complex trigonometric equations?
8. What is the significance of verifying trigonometric identities in calculus?

**Tip:** Always keep the fundamental trigonometric identities in mind. They are essential tools for simplifying and verifying more complex expressions.

Learn how to verify the trigonometric identity Cos²∅(1+tan²∅)=1 with a detailed step-by-step solution. This problem involves trigonometric identities and the Pythagorean identity, essential in trigonometry. Suitable for students in Grades 10-12.

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