To express the given expression $(1 + \frac{1}{\cos^2 x})(1 - \sin^2 x)$ in terms of $\cos x$, follow these steps:

Step 1: Simplify the second factor $(1 - \sin^2 x)$

We know from the Pythagorean identity that: $\sin^2 x + \cos^2 x = 1$ This can be rewritten as: $1 - \sin^2 x = \cos^2 x$ Thus, the expression becomes: $(1 + \frac{1}{\cos^2 x})(\cos^2 x)$

Step 2: Distribute $\cos^2 x$ into the first factor

Now, distribute $\cos^2 x$ across the terms in the first factor: $(1 + \frac{1}{\cos^2 x})(\cos^2 x) = \cos^2 x + \frac{\cos^2 x}{\cos^2 x}$ Simplify the second term: $\cos^2 x + 1$

Final Expression

The expression $(1 + \frac{1}{\cos^2 x})(1 - \sin^2 x)$ simplifies to: $\cos^2 x + 1$

Questions for Further Practice

1. How would you simplify $\frac{1}{1 + \cos^2 x}$?
2. What is the derivative of $\cos^2 x + 1$ with respect to $x$?
3. How does the expression change if we replace $\cos x$ with $\sin x$?
4. Can you simplify $(\cos x + \sin x)^2 + (\cos x - \sin x)^2$?
5. What are the possible values of $x$ if $\cos^2 x + 1 = 2$?

Tip: Always check if identities like the Pythagorean identity can simplify trigonometric expressions before expanding or distributing terms.