Question

If  sinα – sinβ = m and cosα – cosβ = n then cos(α – β) =

(a) [(2 + m² + n²)/2]

(b) [(2 – m² – n²)/2] 

(c) [(m² + n²)/2]

(d) – [(m² + n²)/2]

Answer 1

The correct option (b) [(2 – m² – n²) / 2]

Explanation:

sinα – sinβ = m

cosα – cosβ = n

also cos(α – β) = cosα ∙ cosβ + sinα ∙ sinβ .......(1)

Now m² + n² = sin²α – 2sinα ∙ sinβ + sin²β + cos²α + cos²β – 2cosα ∙ cosβ

m² + n² = 1 + 1 – 2(cosα ∙ cosβ + sinα ∙ sinβ)

∴ [(2 – m² – n²)/2] = cos(α – β) ........ from (1)