Let f(x) be a polynomial such that f(cos theta) = cos(4 theta) for all theta. Find f(x). (This is essentially the same as finding cos(4 theta) in terms of cos theta; we structure the problem this way so that you can answer as a polynomial. Be sure to write your polynomial with the terms in order of decreasing degree.)

Question

Note: The answer is NOT 8cos^4x+8cos^2x+1

Answer 1

cos 4Ø
= 2cos^2 2Ø - 1)
= 2(2cos^2 Ø - 1)(2cos^2 Ø -1) -1
= 2(4cos^4 Ø - 4cos^2 Ø + 1) - 1
= 8cos^4 Ø - 8cos^2 Ø + 2 - 1
= 8cos^4 Ø - 8cos^2 Ø + 1

now let cosØ = x and we have

f(x) = 8x^4 - 8x^2 + 1

Answer 2

OHHH it has to be in terms of x! Thanks!