To verify the identity

$1 + \frac{\tan^2 \theta}{\sec \theta + 1} = \sec \theta,$

we'll start with the left side and simplify it step by step.

Step 1: Start with the left side

$1 + \frac{\tan^2 \theta}{\sec \theta + 1}.$

Step 2: Use the Pythagorean identity

Recall that $\tan^2 \theta = \sec^2 \theta - 1$. Substitute this into the expression: $1 + \frac{\sec^2 \theta - 1}{\sec \theta + 1}.$

Step 3: Combine the fractions

Now we have: $1 + \frac{\sec^2 \theta - 1}{\sec \theta + 1} = \frac{(\sec \theta + 1) + (\sec^2 \theta - 1)}{\sec \theta + 1}.$ Simplifying the numerator: $(\sec \theta + 1) + (\sec^2 \theta - 1) = \sec \theta + \sec^2 \theta.$

Step 4: Simplify further

Now, we can express this as: $\frac{\sec \theta + \sec^2 \theta}{\sec \theta + 1}.$

Step 5: Factor the numerator

Factor out $\sec \theta$ from the numerator: $\sec \theta \cdot \frac{1 + \sec \theta}{\sec \theta + 1}.$

Since $1 + \sec \theta = \sec \theta + 1$, we can cancel these terms: $\sec \theta.$

Conclusion

This shows that $1 + \frac{\tan^2 \theta}{\sec \theta + 1} = \sec \theta.$ Thus, the identity is verified.

If you have any questions or would like further details, feel free to ask!

Follow-Up Questions

1. Can you explain why we use the Pythagorean identity for $\tan^2 \theta$?
2. What are some other identities that could be verified using similar methods?
3. How does factoring help in simplifying trigonometric expressions?
4. What is the significance of verifying trigonometric identities in calculus?
5. Can you give examples of practical applications of trigonometric identities?

Tip

When verifying identities, always look for opportunities to apply Pythagorean identities and factor expressions to simplify your work.