The parameter on which the value of the determinant |{:(1,,a,,a^(2)),(cos (p-d)x ,,cos px,,cos(p+d)x),(sin (p-d)x,,sin px,,sin (p+d)x):}|` does not depend is `a` b. `p` c. `d` d. `x`

Explanation

`"Let " Delta|{:(1,,a,,a^(2)),(cos (p-d)x ,,cos px,,cos(p+d)x),(sin (p-d)x,,sin px,,sin (p+d)x):}|`
Expanding along first row we get, `Delta =1` [cos px sin (p+d)x- cos (p+ d) x sin px]
-a [cos (p-d)x sin (p+d)x - cos (p+d) x sin (p-d)x]
`+a^(2) `[cos (p-d) x sin px - cos px sin (p-d)x]
=sin [(p-d)x -px] -a sin (p-d) x -(p-d)x]
`=a^(2)` sin [px -(p-d) x]
=sin (dx) -a sin (2dx) + `a^(2)` sin (dx)
This is independent of p.