1.2.2 What do you notice concerning the values of $$ \cos (\alpha-\beta) $$ and $$ \cos \alpha-\cos \beta $$ ? (Hint - are the values the same or different?) The Sorme or thoy dern't Pass 1 (1) 1.2.3 What do you notice conceming the values of $$ \cos (\alpha-\beta) $$ and $$ \cos \alpha \cos \beta+\sin \alpha \sin \beta $$ The nurrmbers are increcsiry 1.2.4 What do you notice concerning the values of $$ \cos (\alpha-\beta) $$ and $$ \cos \alpha \cos \beta-\sin \alpha \sin \beta $$ ? The velues are decreasing even futher. 1.2.5 Hence deduce a formula to expand $$ \cos (\alpha-\beta) $$. [17]

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To analyze the relationships between the expressions involving $$ \cos(\alpha - \beta) $$, we will explore each part step by step.

### 1.2.2 Comparison of $$ \cos(\alpha - \beta) $$ and $$ \cos \alpha - \cos \beta $$

1. **Expression for $$ \cos(\alpha - \beta) $$**:
   The cosine of the difference of two angles is given by:
   $$
   \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
   $$

2. **Expression for $$ \cos \alpha - \cos \beta $$**:
   This expression simply represents the difference between the cosines of the two angles.

3. **Comparison**:
   - The two expressions $$ \cos(\alpha - \beta) $$ and $$ \cos \alpha - \cos \beta $$ are generally **not equal**. 
   - The value of $$ \cos(\alpha - \beta) $$ is influenced by both the cosine and sine of the angles, while $$ \cos \alpha - \cos \beta $$ only considers the cosine values.

### 1.2.3 Comparison of $$ \cos(\alpha - \beta) $$ and $$ \cos \alpha \cos \beta + \sin \alpha \sin \beta $$

1. **Expression for $$ \cos(\alpha - \beta) $$**:
   As stated earlier:
   $$
   \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
   $$

2. **Comparison**:
   - Here, $$ \cos(\alpha - \beta) $$ is **equal** to $$ \cos \alpha \cos \beta + \sin \alpha \sin \beta $$.
   - Therefore, they yield the same value.

### 1.2.4 Comparison of $$ \cos(\alpha - \beta) $$ and $$ \cos \alpha \cos \beta - \sin \alpha \sin \beta $$

1. **Expression for $$ \cos(\alpha - \beta) $$**:
   Again, we have:
   $$
   \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
   $$

2. **Expression for $$ \cos \alpha \cos \beta - \sin \alpha \sin \beta $$**:
   This expression represents the difference between the cosine product and the sine product.

3. **Comparison**:
   - The two expressions $$ \cos(\alpha - \beta) $$ and $$ \cos \alpha \cos \beta - \sin \alpha \sin \beta $$ are also **not equal**.
   - In fact, $$ \cos \alpha \cos \beta - \sin \alpha \sin \beta $$ is less than $$ \cos(\alpha - \beta) $$ because it subtracts the sine product instead of adding it.

### 1.2.5 Deduction of the formula for $$ \cos(\alpha - \beta) $$

From the observations made:

- We have established that:
  $$
  \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
  $$
- This is the standard formula for the cosine of the difference of two angles.

Thus, the formula to expand $$ \cos(\alpha - \beta) $$ is:
$$
\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
$$

This concludes the analysis and deduction of the formula.
The formula to expand $$ \cos(\alpha - \beta) $$ is: $$ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta $$

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