if determinant `|{:( cos (theta + phi),,-sin (theta+phi),,cos 2phi),(sin theta,,cos theta,,sin phi),(-cos theta,,sintheta,,cos phi):}|` is

Options

A. non-negative
B. independent of theta
C. independent of `phi`
D. none of these

Answer

A, B

Explanation

Applying `R_(1) to R_(1) +sin phi (R_(2)) +cos phi (R_(3))`
`f(x)= Delta = |{:(0,,0,,cos 2phi+1),(sin0,,cos0,,sinphi),(-cos phi,,sin 0 ,,cos phi):}|`
`=(cos 2phi + 1) (sin^(2) 0+ cos^(2) 0)`
`=(1+cos 2phi)`
Hence `Delta` is independent of 0