Step-by-step explanation:
You want to show cos(4t) +4cos(2t) = 8cos⁴(t).
Double-angle identity
Use the cosine double-angle identity:
cos(2t) = cos²(t) -sin²(t) = cos²(t) -(1 -cos²(t)) = 2cos²(t) -1
Then
cos(4t) = 2cos²(2t) -1 = 2(2cos²(t) -1)² -1 = 2(4cos⁴(t) -4cos²(t) +1) -1
= 8cos⁴(t) -8cos²(t) +1
Application
Substituting these expressions into the left side of the identity of interest, we have ...
cos(4t) +4cos(2t) = 8cos⁴(t) -3 . . . . . . . . . . . . . . . . . . identity to prove
(8cos⁴(t) -8cos²(t) +1) +4(2cos²(t) -1) = 8cos⁴(t) -3 . . . substitute
8cos⁴(t) -8cos²(t) +1 +8cos²(t) -4 = 8cos⁴(t) -3 . . . . . eliminate parentheses
8cos⁴(t) -3 = 8cos⁴(t) -3 . . . . . . combine like terms; Q.E.D.