21.3 Hence, compure number 2.1 .1 and 2.1 .2 and write a conclusion about $$ \cos (\alpha-\beta) $$. (5) $$ \qquad $$ 21.4 Use2.1.3 $$ \cos (\alpha-\beta)=\cos \alpha \cdot \cos \beta+\sin \alpha \cdot \sin \beta] $$ to derive a formula for $$ \cos (\alpha+\beta $$ (Hint: use suitable reduction formula ) $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ (4) 21.5 Use $$ \cos (\alpha-\beta) $$ to derive a formula for $$ \sin (\alpha-\beta) $$. (Hint: use co-function) $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ (3 2.1.6 Use $$ \cos (\alpha-\beta) $$ to derive a formula for $$ \sin (\alpha+\beta) $$. (Hint: use co-function) $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ Ginde 12 Machematics SBA 2025 $$ \sqrt{2} $$

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Let's break down the problem step by step, focusing on the trigonometric identities and their derivations.

### 21.3 Compute $$ \cos(\alpha - \beta) $$

The formula given is:
$$
\cos(\alpha - \beta) = \cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta
$$

To compute $$ \cos(\alpha - \beta) $$, we need to evaluate the expressions for $$ \cos \alpha $$, $$ \cos \beta $$, $$ \sin \alpha $$, and $$ \sin \beta $$. However, since specific values for $$ \alpha $$ and $$ \beta $$ are not provided, we can only state the formula as is.

### Conclusion about $$ \cos(\alpha - \beta) $$

The conclusion we can draw from the formula is that the cosine of the difference of two angles can be expressed as the product of their cosines plus the product of their sines. This identity is useful in simplifying expressions involving the cosine of angle differences.

### 21.4 Derive a formula for $$ \cos(\alpha + \beta) $$

To derive $$ \cos(\alpha + \beta) $$ using the reduction formula, we can use the identity for $$ \cos(\alpha - \beta) $$ and replace $$ \beta $$ with $$-\beta$$:

$$
\cos(\alpha + \beta) = \cos(\alpha - (-\beta)) = \cos \alpha \cdot \cos(-\beta) + \sin \alpha \cdot \sin(-\beta)
$$

Using the properties of cosine and sine:
$$
\cos(-\beta) = \cos \beta \quad \text{and} \quad \sin(-\beta) = -\sin \beta
$$

Substituting these into the equation gives:
$$
\cos(\alpha + \beta) = \cos \alpha \cdot \cos \beta - \sin \alpha \cdot \sin \beta
$$

### 21.5 Derive a formula for $$ \sin(\alpha - \beta) $$

Using the co-function identity, we know that:
$$
\sin(\alpha - \beta) = \cos\left(\frac{\pi}{2} - (\alpha - \beta)\right) = \cos\left(\frac{\pi}{2} - \alpha + \beta\right)
$$

Using the cosine of a sum:
$$
\sin(\alpha - \beta) = \cos\left(\frac{\pi}{2} - \alpha\right) \cdot \cos(\beta) + \sin\left(\frac{\pi}{2} - \alpha\right) \cdot \sin(\beta)
$$

This simplifies to:
$$
\sin(\alpha - \beta) = \sin \alpha \cdot \cos \beta - \cos \alpha \cdot \sin \beta
$$

### 21.6 Derive a formula for $$ \sin(\alpha + \beta) $$

Using the co-function identity again:
$$
\sin(\alpha + \beta) = \cos\left(\frac{\pi}{2} - (\alpha + \beta)\right) = \cos\left(\frac{\pi}{2} - \alpha - \beta\right)
$$

Using the cosine of a sum:
$$
\sin(\alpha + \beta) = \cos\left(\frac{\pi}{2} - \alpha\right) \cdot \cos(\beta) + \sin\left(\frac{\pi}{2} - \alpha\right) \cdot \sin(\beta)
$$

This simplifies to:
$$
\sin(\alpha + \beta) = \sin \alpha \cdot \cos \beta + \cos \alpha \cdot \sin \beta
$$

### Summary of Results

1. **Cosine of the difference**:
   $$
   \cos(\alpha - \beta) = \cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta
   $$

2. **Cosine of the sum**:
   $$
   \cos(\alpha + \beta) = \cos \alpha \cdot \cos \beta - \sin \alpha \cdot \sin \beta
   $$

3. **Sine of the difference**:
   $$
   \sin(\alpha - \beta) = \sin \alpha \cdot \cos \beta - \cos \alpha \cdot \sin \beta
   $$

4. **Sine of the sum**:
   $$
   \sin(\alpha + \beta) = \sin \alpha \cdot \cos \beta + \cos \alpha \cdot \sin \beta
   $$

These identities are fundamental in trigonometry and are widely used in various applications, including physics and engineering.
Here are the derived formulas: 1. **Cosine of the difference**: $$ \cos(\alpha - \beta) = \cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta $$ 2. **Cosine of the sum**: $$ \cos(\alpha + \beta) = \cos \alpha \cdot \cos \beta - \sin \alpha \cdot \sin \beta $$ 3. **Sine of the difference**: $$ \sin(\alpha - \beta) = \sin \alpha \cdot \cos \beta - \cos \alpha \cdot \sin \beta $$ 4. **Sine of the sum**: $$ \sin(\alpha + \beta) = \sin \alpha \cdot \cos \beta + \cos \alpha \cdot \sin \beta $$ These identities help in simplifying trigonometric expressions involving angle differences and sums.

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