Is cos(θ)-cos^3(θ)=sin^2(cos(θ)) ?

The answer to whether cos(θ)-cos^3(θ)=sin^2(cos(θ)) is False

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prove cos(θ)-cos^3(θ)=sin^2(cos(θ))

prove

cos (θ) - cos^{3} (θ) = sin^{2} (cos (θ))

False

Solution steps

cos (θ) - cos^{3} (θ) = sin^{2} (cos (θ))

Show that the two sides are not equal

For

cos (θ) - cos^{3} (θ) = sin^{2} (cos (θ))

plug in

θ = 0

cos (0) - cos^{3} (0) = 0

cos (0) - cos^{3} (0)

Use the following trivial identity:

cos (0) = 1

cos (0)

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 1

= 1 - 1^{3}

Simplify

= 0

sin^{2} (cos (0)) = 0.70807 \dots

sin^{2} (cos (0))

Refine to a decimal form

= 0.70807 \dots

The two sides are not equal

\Rightarrow False

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Is cos(θ)-cos^3(θ)=sin^2(cos(θ)) ?
The answer to whether cos(θ)-cos^3(θ)=sin^2(cos(θ)) is False