The expression cos⁴ θ in terms of the first power of cosine is [ 3 + 2cos 2θ + cos 4θ]/8.
The power-reducing formula, for cosine, is,
cos² θ = (1/2)[1 + cos 2θ].
In the question, we are asked to use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine cos⁴ θ.
We can do it as follows:
cos⁴ θ
= (cos² θ)²
= {(1/2)[1 + cos 2θ]}²
= (1/4)[1 + cos 2θ]²
= (1/4)(1 + 2cos 2θ + cos² 2θ] {Using (a + b)² = a² + 2ab + b²}
= 1/4 + (1/2)cos 2θ + (1/4)(cos ² 2θ)
= 1/4 + (1/2)cos 2θ + (1/4)(1/2)[1 + cos 4θ]
= 1/4 + cos 2θ/4 + 1/8 + cos 4θ/8
= 3/8 + cos 2θ/4 + cos 4θ/8
= [ 3 + 2cos 2θ + cos 4θ]/8.
Thus, the expression cos⁴ θ in terms of the first power of cosine is [ 3 + 2cos 2θ + cos 4θ]/8.
Learn more about reducing trigonometric powers at
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