2.1. Use $$ \cos (\alpha-\beta) $$ to derive a formula for $$ \sin (\alpha+\beta) $$. (Ifint: use co-function)

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Sure, let's derive the formula for $$ \sin(\alpha + \beta) $$ using $$ \cos(\alpha - \beta) $$ and the co-function identities.

### Step 1: Use the Co-Function Identity

The co-function identity relates sine and cosine functions with complementary angles. Specifically:
$$
\sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right)
$$

Apply this identity to $$ \sin(\alpha + \beta) $$:
$$
\sin(\alpha + \beta) = \cos\left(\frac{\pi}{2} - (\alpha + \beta)\right) = \cos\left(\frac{\pi}{2} - \alpha - \beta\right)
$$

### Step 2: Express $$ \cos\left(\frac{\pi}{2} - \alpha - \beta\right) $$ as a Difference

Notice that:
$$
\frac{\pi}{2} - \alpha - \beta = \left(\frac{\pi}{2} - \alpha\right) - \beta
$$
Let’s denote:
$$
u = \frac{\pi}{2} - \alpha \quad \text{and} \quad v = \beta
$$
So,
$$
\frac{\pi}{2} - \alpha - \beta = u - v
$$

### Step 3: Apply the Cosine of a Difference Formula

Recall the cosine difference identity:
$$
\cos(u - v) = \cos u \cos v + \sin u \sin v
$$

Apply this to our expression:
$$
\cos\left(\frac{\pi}{2} - \alpha - \beta\right) = \cos(u - v) = \cos u \cos v + \sin u \sin v
$$

### Step 4: Substitute Back $$ u $$ and $$ v $$

Substitute $$ u = \frac{\pi}{2} - \alpha $$ and $$ v = \beta $$:
$$
\cos u = \cos\left(\frac{\pi}{2} - \alpha\right) = \sin \alpha \quad \text{(using the co-function identity)}
$$
$$
\sin u = \sin\left(\frac{\pi}{2} - \alpha\right) = \cos \alpha \quad \text{(using the co-function identity)}
$$
$$
\cos v = \cos \beta
$$
$$
\sin v = \sin \beta
$$

Substitute these into the equation:
$$
\cos(u - v) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
$$

### Step 5: Combine the Results

Recall that:
$$
\sin(\alpha + \beta) = \cos\left(\frac{\pi}{2} - \alpha - \beta\right) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
$$

### Final Formula

Thus, we have derived the sine addition formula:
$$
\boxed{\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta}
$$

### Summary

By leveraging the co-function identity and the cosine of a difference formula, we've successfully derived the formula for $$ \sin(\alpha + \beta) $$ in terms of trigonometric functions of $$ \alpha $$ and $$ \beta $$.
The formula for $$ \sin(\alpha + \beta) $$ is: $$ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta $$

2.1. Use \( \cos (\alpha-\beta) \) to derive a formula for \( \sin (\alpha+\beta) \). (Ifint: use co-function)

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