The number of values of x which satify the equation `cos^(2)x =1` and `x^(2) le 20` are _________

Answer

To solve the equation \( \cos^2 x = 1 \) under the constraint \( x^2 \leq 20 \), we will follow these steps:

### Step 1: Solve the equation \( \cos^2 x = 1 \)

We start with the equation:
\[ \cos^2 x = 1 \]
This can be rewritten as:
\[ \cos^2 x - 1 = 0 \]
Using the difference of squares, we factor it:
\[ (\cos x - 1)(\cos x + 1) = 0 \]
This gives us two equations to solve:
1. \( \cos x - 1 = 0 \) which leads to \( \cos x = 1 \)
2. \( \cos x + 1 = 0 \) which leads to \( \cos x = -1 \)

### Step 2: Find the values of \( x \)

For \( \cos x = 1 \):
- The general solution is:
\[ x = 2n\pi \quad (n \in \mathbb{Z}) \]
The specific value in the range of \( x \) satisfying \( x^2 \leq 20 \) is:
\[ x = 0 \]

For \( \cos x = -1 \):
- The general solution is:
\[ x = (2n + 1)\pi \quad (n \in \mathbb{Z}) \]
The specific values in the range of \( x \) satisfying \( x^2 \leq 20 \) are:
\[ x = \pi \quad \text{and} \quad x = -\pi \]

### Step 3: Determine the constraint \( x^2 \leq 20 \)

Now we need to find the range of \( x \) that satisfies the constraint:
\[ x^2 \leq 20 \]
This can be rewritten as:
\[ -\sqrt{20} \leq x \leq \sqrt{20} \]
Calculating \( \sqrt{20} \):
\[ \sqrt{20} = 2\sqrt{5} \]
Thus, the range is:
\[ -2\sqrt{5} \leq x \leq 2\sqrt{5} \]

### Step 4: Identify the valid solutions

Now we check the values of \( x \) we found earlier:
- From \( \cos x = 1 \): \( x = 0 \)
- From \( \cos x = -1 \): \( x = \pi \) and \( x = -\pi \)

We need to check if these values fall within the range \( -2\sqrt{5} \leq x \leq 2\sqrt{5} \):
- \( 0 \) is in the range.
- \( \pi \approx 3.14 \) is in the range since \( 2\sqrt{5} \approx 4.47 \).
- \( -\pi \approx -3.14 \) is also in the range since \( -2\sqrt{5} \approx -4.47 \).

### Conclusion

Thus, the values of \( x \) that satisfy both \( \cos^2 x = 1 \) and \( x^2 \leq 20 \) are:
1. \( x = 0 \)
2. \( x = \pi \)
3. \( x = -\pi \)

Therefore, the number of values of \( x \) that satisfy the equation is:
\[ \boxed{3} \]