Is cos(θ)-cos^3(θ)=sin^2(θ)cos(θ) ?

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

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prove cos(θ)-cos^3(θ)=sin^2(θ)cos(θ)

prove

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

True

Solution steps

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

Manipulating left side

cos (θ) - cos^{3} (θ)

Factor

cos (θ) - cos^{3} (θ) : cos (θ) (1 - cos^{2} (θ))

cos (θ) - cos^{3} (θ)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

cos^{3} (θ) = cos (θ) cos^{2} (θ)

= cos (θ) - cos (θ) cos^{2} (θ)

Factor out common term

cos (θ)

= cos (θ) (1 - cos^{2} (θ))

= (1 - cos^{2} (θ)) cos (θ)

Manipulating right side

sin^{2} (θ) cos (θ)

Rewrite using trig identities

sin^{2} (θ) cos (θ)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= (1 - cos^{2} (θ)) cos (θ)

= (1 - cos^{2} (θ)) cos (θ)

We showed that the two sides could take the same form

\Rightarrow True

\frac{cos ( 4 x ) + 1}{sin ( 4 x )} = cot (2 x)

2 sin (θ) cos (θ) = tan (2 θ) cos (2 θ)

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

sin (12 x) = 2 sin (6 x) cos (6 x)

\frac{sin ( t )}{csc ( t )} = 1 - \frac{cos ( t )}{sec ( t )}

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