Question
Show that Cos 4¤=8cos^4¤ - 8cos^2 + 1
Answers
GPT-4
The formula that we can use is the double-angle formula, which states that cos(2x) = 2cos^2(x) - 1. We can use this formula twice to get the cos(4x) in terms of cos(x).
First, apply the double-angle formula for x:
cos(2x) = 2cos^2(x) - 1 ...(1)
Then, apply the double-angle formula again, but now for 2x:
cos(4x) = 2cos^2(2x) - 1 ...(2)
Substituting equation (1) into equation (2)
cos(4x) = 2[2cos^2(x) - 1]^2 - 1
= 2[4cos^4(x) - 4cos^2(x) + 1] - 1
= 8cos^4(x) - 8cos^2(x) + 2 - 1
= 8cos^4(x) - 8cos^2(x) + 1
Thus, cos(4x) = 8cos^4(x) - 8cos^2(x) + 1.
First, apply the double-angle formula for x:
cos(2x) = 2cos^2(x) - 1 ...(1)
Then, apply the double-angle formula again, but now for 2x:
cos(4x) = 2cos^2(2x) - 1 ...(2)
Substituting equation (1) into equation (2)
cos(4x) = 2[2cos^2(x) - 1]^2 - 1
= 2[4cos^4(x) - 4cos^2(x) + 1] - 1
= 8cos^4(x) - 8cos^2(x) + 2 - 1
= 8cos^4(x) - 8cos^2(x) + 1
Thus, cos(4x) = 8cos^4(x) - 8cos^2(x) + 1.