Write the expression “(1+1/cos^2x)(1-sin^2x)” in terms of cos x

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.

Learn how to express the trigonometric expression (1 + 1/cos^2x)(1 - sin^2x) in terms of cos x with detailed step-by-step solutions. This problem involves applying the Pythagorean identity and simplifying trigonometric expressions. Suitable for advanced high school students in Grades 11-12.

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