<Question>
Simplify $\frac{1}{1+\sqrt{2}}\cdot\frac{1}{1-\sqrt{2}}$.
</Question>

<Plan_1>
Okay, so I need to simplify the expression (1/(1 + √2)) * (1/(1 - √2)). Hmm, let's see. First, I remember that when you multiply two fractions, you multiply the numerators together and the denominators together.

So, the numerator here would be 1 * 1, which is 1. The denominator would be (1 + √2)(1 - √2).

Wait, the denominator is a product of two binomials that are conjugates. Conjugate pairs... right? Because one has a plus and the other has a minus with the same terms. I think when you multiply conjugates, it's a difference of squares. So, (a + b)(a - b) = a² - b². Let me confirm that.

Yes, that formula should work here.

So applying that to the denominator: (1 + √2)(1 - √2) = 1² - (√2)². Calculating each part, 1 squared is 1, and √2 squared is 2. So, 1 - 2 = -1. Therefore, the denominator simplifies to -1.

Putting it all together, the original expression becomes 1 divided by -1, which is just -1.

Wait, let me double-check.

The numerators multiply to 1*1=1, denominators multiply to (1 + √2)(1 - √2) = -1. So, 1/-1 is indeed -1.

But let me make sure I didn't skip any steps or make a mistake in multiplying the denominators. Let me expand (1 + √2)(1 - √2) manually.

Using distributive property: 1*1 = 1, 1*(-√2) = -√2, √2*1 = √2, and √2*(-√2) = - (√2)^2. So combining those terms: 1 - √2 + √2 - (√2)^2. The -√2 and +√2 cancel each other out, leaving 1 - (√2)^2. Which is 1 - 2 = -1.

Wait a second, but maybe the question expects the answer in a different form? Sometimes when simplifying expressions with radicals, you have to rationalize the denominator.

But in this case, after multiplying, the denominator becomes a rational number (-1), so there's no radical left. Therefore, the answer is just -1.

Alternatively, maybe there's another approach to verify. Let me compute each fraction separately first, then multiply them. So, 1/(1 + √2). To rationalize the denominator, I can multiply numerator and denominator by the conjugate (1 - √2).

<prompt_1.1> Rationalize each fraction separately and multiply them as per the above plan. </prompt_1.1>
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Another way to think about it: since the original expression is the product of a number and its reciprocal? Wait, no, the denominators are conjugates. So maybe there's a property here.

<prompt_1.2> Consider the properties of conjugates and their reciprocals to verify the result. </prompt_1.2>

Just to recap:

Original expression: [1/(1 + √2)] * [1/(1 - √2)]

Multiply numerators: 1*1 = 1

Multiply denominators: (1 + √2)(1 - √2) = 1 - (√2)^2 = 1 - 2 = -1

Thus, 1 / (-1) = -1.

All steps are correct, and multiple methods confirm the same result.
</Plan_1>

<Execution_1>
<execution_1.1>
Let's try that: 

1/(1 + √2) * (1 - √2)/(1 - √2) = [1*(1 - √2)] / [(1 + √2)(1 - √2)]. As before, the denominator is -1, so this becomes (1 - √2)/(-1) = - (1 - √2) = √2 - 1. 

Similarly, the second fraction is 1/(1 - √2). To rationalize this, multiply numerator and denominator by (1 + √2):

1/(1 - √2) * (1 + √2)/(1 + √2) = [1*(1 + √2)] / [(1 - √2)(1 + √2)] = (1 + √2)/(-1) = - (1 + √2) = -1 - √2.

Now, if I multiply the two rationalized forms: (√2 - 1) * (-1 - √2). Let's compute this:

First, expand the multiplication using FOIL:

√2 * (-1) = -√2

√2 * (-√2) = - (√2 * √2) = -2

-1 * (-1) = 1

-1 * (-√2) = √2

Combine all terms: -√2 - 2 + 1 + √2. 

The -√2 and +√2 cancel out, leaving -2 + 1 = -1. 

So that also results in -1.
</execution_1.1>

<execution_1.2>
But regardless, through both multiplying directly and rationalizing each term first, we arrive at the same answer. 

Alternatively, recognizing that 1/(1 + √2) and 1/(1 - √2) are reciprocals of conjugates. But their product would be 1 divided by the product of (1 + √2)(1 - √2), which again is 1/-1, so -1. 

Therefore, all approaches converge on -1.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, so both methods lead to the same result.

That confirms that the original expression simplifies to -1.
</Plan_2>

<Execution_2>
No prompt to execute.
</Execution_2>

<Final_answer>
Therefore, the simplified expression is $\boxed{-1}$.
</Final_answer>