Question

If `p=sin(A-B)sin(C-D),q=sin(B-C)sin(A-D)`, `r=sin(C-A)sin(B-D)` then (a) `p+q-r=0` (b) `p+q+r=0` `p-q+r=0` (d) `p^3+q^3+r^3=3p q r`

Answer 1

Here, we will use,
`2sinCsinD = cos(C-D) - cos(C+D)`
Now, using the above identity,
`2p = cos(A-B-C+D) - cos(A-B+C-D)->(1)`
`2q = cos(B-C-A+D) - cos(B-C+A-D)->(2)`
`2r = cos(C-A-B+D) - cos(C-A+B-D)->(3)`
Now, adding (1),(2) and (3),
`2(p+q+r) = cos(A-B-C+D) - cos(A-B+C-D)+cos(B-C-A+D) - cos(B-C+A-D)+cos(C-A-B+D) - cos(C-A+B-D)`
`=>2(p+q+r) = 0`
`=>p+q+r = 0`
`:.` Option `b` is the correct option.