Let $z$ and $w$ be complex numbers such that $|z| = |w| = 1$ and $zw \ne -1.$

(a) Prove that $\overline{z} = \frac{1}{z}$ and $\overline{w} = \frac{1}{w}.$

To prove that $\overline{z} = \frac{1}{z}$ and $\overline{w} = \frac{1}{w}$, let's break the problem down step by step for $z$. The same reasoning will apply for $w$.

### Step 1: Use the property $|z| = 1$

Given that $|z| = 1$, recall that the modulus of a complex number $z = a + bi$, where $a$ and $b$ are real, is defined as:

\[
|z| = \sqrt{a^2 + b^2}
\]

However, since $|z| = 1$, this implies:

\[
\sqrt{a^2 + b^2} = 1 \quad \Rightarrow \quad a^2 + b^2 = 1
\]

This tells us that the complex number $z$ lies on the unit circle in the complex plane.

### Step 2: Conjugate of $z$

The complex conjugate of $z = a + bi$ is:

\[
\overline{z} = a - bi
\]

### Step 3: Inverse of $z$

The inverse of $z = a + bi$ is given by:

\[
\frac{1}{z} = \frac{1}{a + bi}
\]

To simplify $\frac{1}{a + bi}$, we multiply both the numerator and the denominator by the complex conjugate of $a + bi$, which is $a - bi$. This yields:

\[
\frac{1}{z} = \frac{a - bi}{(a + bi)(a - bi)} = \frac{a - bi}{a^2 + b^2}
\]

Since we know $a^2 + b^2 = 1$ (from Step 1), the expression simplifies to:

\[
\frac{1}{z} = a - bi = \overline{z}
\]

Thus, we have shown that:

\[
\overline{z} = \frac{1}{z}
\]

### Step 4: Apply the same reasoning to $w$

Since $|w| = 1$, we can follow exactly the same steps as above to show that:

\[
\overline{w} = \frac{1}{w}
\]

### Conclusion

We have proven that:

\[
\overline{z} = \frac{1}{z} \quad \text{and} \quad \overline{w} = \frac{1}{w}
\]

Would you like further clarification or details on any part? 

Here are some related questions:
1. How does the unit circle help in understanding complex numbers?
2. Can you show a general proof for other moduli $ |z| \neq 1 $?
3. How would this change if $ z $ and $ w $ were not on the unit circle?
4. What other properties of complex numbers can be derived from $ |z| = 1 $?
5. How can this proof be extended to other forms of $z$, such as polar form?

**Tip**: When dealing with complex numbers on the unit circle, it's often helpful to use their polar form, $ z = e^{i\theta} $, where computations involving modulus and conjugates become more straightforward.

Let $z$ and $w$ be complex numbers such that $|z| = |w| = 1$ and $zw \ne -1.$ (a) Prove that $\overline{z} = \frac{1}{z}$ and $\overline{w} = \frac{1}{w}.$

Explore the proof that for complex numbers on the unit circle, their conjugate equals their inverse. This solution covers the modulus, conjugates, and inverses of complex numbers, suitable for advanced high school or undergraduate students.

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