於是cos(4x) = 2cos?(2x)-1 = 2(2cos?(x)-1)?-1 = 8cos(x)^4-8cos?(x)+1.
又sin(2x) = 2sin(x)cos(x), sin(4x) = 2sin(2x)cos(2x) = 4sin(x)cos(x)(2cos?(x)-1).
cos(5x) = cos(4x)cos(x)-sin(4x)sin(x) = 8cos(x)^5-8cos?(x)+cos(x)-4sin?(x)cos(x)(2cos?(x)-1)
= 8cos(x)^5-8cos?(x)+cos(x)-4(1-cos?(x))(2cos?(x)-1)cos(x)
= 16cos(x)^5-20cos?(x)+5cos(x).
因此P5(x) = 16x^5-20x?+5x.
易見P5(x)在[-1,1]上的值域 = P5(cos(x))在R上的值域.
而P5(cos(x)) = cos(5x), 因此值域為[-1,1].
故P5(x)在[-1,1]上的值域為[-1,1].