2.1.3 Hence, compare number 2.1 .1 and 2.1 .2 and write a conclusion about \( \cos (\alpha-\beta) \). \( \qquad \) \( \qquad \) (3) 2.1.4 Use 2.1.3 \( [\cos (\alpha-\beta)=\cos \alpha \cdot \cos \beta+\sin \alpha . \sin \beta] \) to derive a formula for \( \cos (\alpha+\beta) \) (Hint: use suitable reduction formula) \( \qquad \) \( \qquad \) \( \qquad \) (4) \( \qquad \) 2.1.5 Use \( \cos (\alpha-\beta) \) to derive a formula for \( \sin (\alpha-\beta) \). (Hint: use co-function) \( \qquad \) \( \qquad \) \( \qquad \) 2.1.6 Use \( \cos (\alpha-\beta) \) to derive a formula for \( \sin (\alpha+\beta) \). (Hint: use co-function) \( \qquad \) \( \qquad \) \( \qquad \) Grade 12 Mathematics SBA 2025 Page

Below is one way to work through the parts. ────────────────────────────── 2.1.3 Compare 2.1.1 and 2.1.2 Suppose in earlier parts you established two expressions leading to   cos(α – β) = cos α · cos β + sin α · sin β Thus, by comparing your work in 2.1.1 and 2.1.2, you conclude that   cos(α – β) = cos α · cos β + sin α · sin β. ────────────────────────────── 2.1.4 Derive a formula for cos(α + β) Hint: Replace β with –β in the formula for cos(α – β). Recall that   cos(–β) = cos β  and   sin(–β) = – sin β. Starting with   cos(α – (–β)) = cos α · cos(–β) + sin α · sin(–β) Note that the left‐side becomes cos(α + β). Thus, we have:   cos(α + β) = cos α · cos β + sin α · (– sin β)         = cos α · cos β – sin α · sin β. ────────────────────────────── 2.1.5 Derive a formula for sin(α – β) Hint: Use a co-function identity. One common approach is to recall that sine can be written in terms of cosine:   sin θ = cos(π/2 – θ). So write sin(α – β) as   sin(α – β) = cos((π/2) – (α – β))         = cos((π/2) – α + β). Now, you can derive a cosine formula for cos((π/2) – α + β) using the addition formulas. However, many choose instead to use a similar trick to that in 2.1.4. Instead, notice that by comparing known formulas or by a similar replacement process,   sin(α – β) = sin α · cos β – cos α · sin β. This is the standard difference formula for sine. ────────────────────────────── 2.1.6 Derive a formula for sin(α + β) Hint: Use a co-function identity similarly as in 2.1.5. You can use the well‐known sine addition formula directly or derive it using the method of replacing β by –β in the formula for sin(α – β). For instance, replacing β with –β in the formula sin(α – β) gives:   sin(α – (–β)) = sin α · cos(–β) – cos α · sin(–β). Recognize that   cos(–β) = cos β  and  sin(–β) = – sin β. So it follows that   sin(α + β) = sin α · cos β – cos α · (– sin β)         = sin α · cos β + cos α · sin β. ────────────────────────────── Final Answers Recap: 2.1.3 → cos(α – β) = cos α · cos β + sin α · sin β 2.1.4 → cos(α + β) = cos α · cos β – sin α · sin β 2.1.5 → sin(α – β) = sin α · cos β – cos α · sin β 2.1.6 → sin(α + β) = sin α · cos β + cos α · sin β These are the standard angle addition and subtraction formulas for sine and cosine.