Write an equivalent expressions for 8cos^4 (x) that does not contain powers of trigonometric functions greater than 1

Question

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Answer 1

8•cos⁴(x)

=> 8•cos²(x)•cos²(x)

Using Half Angle Formula:

cos²θ = (1+cos 2θ)/2

=> 8•[(1+cos 2x)/2][(1+cos 2x)/2]

=> 8•(1+2•cos 2x+cos² 2x)/4

=> 2•(1+2•cos 2x+cos² 2x)

=>2+4•cos 2x+2•cos² 2x

Using Half-Angle Formula again:

=>2+4•cos 2x+2•[(1+ cos 4x)/2]

=>2+4•cos 2x+1+ cos 4x

Combine like terms:

=>3+4•cos 2x+cos 4x