Show that, d/dz( cosz)=-sinz, where z= x+ iy

To prove that:

\[
\frac{d}{dz} (\cos z) = -\sin z
\]

where \( z = x + iy \) (with \( x \) and \( y \) being real), we need to approach this using the **Cauchy-Riemann equations** and the complex definition of the trigonometric functions.

### Step 1: Express \( \cos z \) and \( \sin z \) in terms of complex exponentials.

From Euler's formulas, we know:

\[
\cos z = \frac{e^{iz} + e^{-iz}}{2}, \quad \sin z = \frac{e^{iz} - e^{-iz}}{2i}
\]

Now, differentiate \( \cos z \) with respect to \( z \).

### Step 2: Differentiate using the exponential form.

Differentiate \( \cos z \) using the exponential definition:

\[
\frac{d}{dz} \left( \frac{e^{iz} + e^{-iz}}{2} \right)
\]

Apply the chain rule to each exponential term:

\[
\frac{1}{2} \left( \frac{d}{dz} e^{iz} + \frac{d}{dz} e^{-iz} \right)
\]

The derivative of \( e^{iz} \) with respect to \( z \) is \( i e^{iz} \), and the derivative of \( e^{-iz} \) is \( -i e^{-iz} \). So, we get:

\[
\frac{1}{2} \left( i e^{iz} - i e^{-iz} \right)
\]

### Step 3: Simplify the result.

Factor out \( i \) and simplify:

\[
\frac{i}{2} \left( e^{iz} - e^{-iz} \right) = -\frac{e^{iz} - e^{-iz}}{2i}
\]

But we recognize that this is just the definition of \( \sin z \):

\[
-\frac{e^{iz} - e^{-iz}}{2i} = -\sin z
\]

### Conclusion:
Thus, we have shown that:

\[
\frac{d}{dz} (\cos z) = -\sin z
\]

Would you like any further clarifications or details? Here are some related questions to explore:

1. What are the Cauchy-Riemann equations and how do they relate to this proof?
2. How does the differentiation of complex functions differ from real functions?
3. Can you show the differentiation of \( \sin z \) with respect to \( z \)?
4. How does the chain rule apply in the context of complex analysis?
5. How can we apply similar methods to differentiate other complex functions like \( e^z \)?

**Tip:** When working with complex functions, it's often useful to express them in terms of exponential functions for easier differentiation and manipulation.

Learn how to differentiate the complex trigonometric function cos(z) with respect to z = x + iy, using Euler's formula and the Cauchy-Riemann equations. This detailed proof shows why d/dz(cos(z)) = -sin(z), perfect for students studying complex analysis at the undergraduate level.

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