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Step-by-step explanation:
1cos4(θ)−cos4(θ)=1−2cos2(θ)
Multiply both sides by cos4(θ)
1−cos8(θ)=cos4(θ)−2cos6(θ)
Let u=cos2(θ)
1−u4=u2−2u3
u4−2u3+u2−1=0(1)
Since this is a degree 4 equation, it can have at most 4 solutions. That means that cos2(θ) can have at most 4 values that satisfy(1). Since only 4 values can be satisfied at max, and |R|≫4, your original equation is not true for the vast majority of θ values.
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