Without expanding evaluate the determinant `|sinalphacosalphasin(alpha+delta)sinbetacosbetasin(beta+delta)singammacosgammasin(gamma+delta)|` .

Options

A. 0
B. `sin alpha sin beta sin gamma`
C. `cos alpha cos beta cos gamma`
D. none of these

Answer

A

Explanation

We have,
`Delta = |(sin alpha,cos alpha,sin alpha cos delta + cos alpha sin delta),(sin beta,cos beta,sin beta cos delta + cos beta sin delta),(sin gamma,cos gamma,sin gamma cos delta + cos gamma sin delta)|`
`rArr Dlta = |(sin alpha,cos alpha,0),(sin beta,cos beta,0),(sin gamma,cos gamma,0)| " " [("Applying " C_(3) rarr C_(3) - cos delta C_(1)),(" "- sin deltaC_(2))]`
`rArr Delta = 0 [ :' C_(3) " consists of all zeros"]`