Given:
(cosx−cos3x)(sin8x+sin2x)(sin5x−sinx)(cos4x−cos6x)
Use the formula:
sina+sinb=2sin(a+b2)cos(a−b2)
sina−sinb=2cos(a+b2)sin(a−b2)
cosa−cosb=−2sin(a+b2)sin(a−b2)
(cosx−cos3x)(sin8x+sin2x)(sin5x−sinx)(cos4x−cos6x)=(−2sin4x2sin−2x2)(2sin10x2cos6x2)(2cos6x2sin4x2)(−2sin10x2sin−2x2)
=(2sin2xsinx)(2sin5xcos3x)(2cos3xsin2x)(2sin5xsinx)
=1
Thus, the value is 1.