+322 Use the power-reducing formulas as many times as possible to rewrite the expression in terms of the first power of the cosine.
7 cos4 x
+6251 \(\text{I assume you mean }7\cos^4(x)\)
\(\cos^4(x) =\left( \cos^2(x)\right)^2 = \\ \left(\dfrac{\cos(2x)+1}{2}\right)^2 = \\ \dfrac 1 4 + \dfrac 1 2 \cos(2x) + \dfrac 1 4 \cos^2(2x)\\ \text{repeat the process on the term on the right}\\ \dfrac 1 4 \cos^2(2x) = \dfrac 1 8 (1 + \cos(4x)) \\ \text{so combining everything}\\ 7\cos^4(x) = \dfrac{7}{8} (4 \cos (2 x)+\cos (4 x)+3)\)
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