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

Options

A. `p+q-r=0`
B. `p+q+r=0`
C. `p-q+r=0`
D. `p^(3)+q^(3)+r^(3)=3pqr`

Answer

B, D

Explanation

` 2p = cos ( A-B -C +D) - cos (A -B + C-D)`
`rArr " " 2q = cos (B-C-A +D)-cos(B-C +A-D)`
`rArr " "2r = cos(C -A -B +D) -cos (C-A +B-D)`
`rArr " " 2p + 2q + 2r = 0`
`" " p+q + r =0 and p^(3) +q^(3) +r^(3)= 3pqr`