What is value of \(\begin{vmatrix}1 & cosx-sinx & cosx + sinx \\1 & cosy-siny & cosy + siny \\1 & cosz-sinz & cosz + sinz \end {vmatrix}\)?

Let, a = cosx, b = cosy, c = cosz, p =sinx, q = siny and r = sinz
So, \(\begin{vmatrix}1 & a – p & a + p \\1 & b – q & b + q \\1 & c – r & c + r \end {vmatrix}\)
Making C₃ = C₃ + C₂
= \(\begin{vmatrix}1 & a – p & 2a \\1 & b – q & 2b \\1 & c – r & 2c \end {vmatrix}\)

= 2\(\begin{vmatrix}1 & a – p & a \\1 & b – q & b \\1 & c – r & c \end {vmatrix}\)
Making C₂ = C₂ – C₃
= -2\(\begin{vmatrix}1 & p & a \\1 & q & b \\1 & r & c \end {vmatrix}\)
Interchanging 2^nd and 3^rd column, we get,
2\(\begin{vmatrix}1 & a & p \\1 & b & q \\1 & c & r \end {vmatrix}\)
= 2\(\begin{vmatrix}1 & cosx & sinx \\1 & cosy & siny \\1 & cosz & sinz \end {vmatrix}\)