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Show that cos4(t)+4cos2(t)=8cos^4(t)-3​

1 Answer

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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.

User Juan M
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