{cos α + cos β} / {sin α + sin β} + {sin α - sin β} / {cos α - cos β} =

Question

\(\frac{\cos \alpha + \cos \beta}{\sin \alpha + \sin \beta} + \frac{\sin \alpha - \sin \beta}{\cos \alpha - \cos \beta} =\)

(a) 1

(b) 2

(c) √2

(d) 0

Answer 1

Correct option is (d) 0

\(\frac{\cos \alpha + \cos \beta}{\sin \alpha + \sin \beta} + \frac{\sin \alpha - \sin \beta}{\cos \alpha - \cos \beta} \)

\(=\frac{(\cos \alpha + \cos \beta)(\cos \alpha - \cos \beta) + (\sin \alpha + \sin \beta) (\sin \alpha - \sin \beta)}{(\sin \alpha + \sin \beta)(\cos \alpha - \cos \beta)}\)

\(=\frac{(\cos^2\alpha - \cos^2\beta) + (\sin^2\alpha - \sin^2\beta)}{(\sin \alpha + \sin \beta)(\cos \alpha - \cos \beta)}\)

\(=\frac{\cos^2\alpha+ \cos^2\beta + \sin^2\alpha - \sin^2\beta}{(\sin \alpha + \sin \beta)(\cos \alpha - \cos \beta)}\)

\(=\frac{1- 1}{(\sin \alpha + \sin \beta)(\cos \alpha - \cos \beta)}\)

\(=\frac{0}{(\sin \alpha + \sin \beta)(\cos \alpha - \cos \beta)}\)

\(=0\)