Is cos(θ)-cos(θ)sin^2(θ)=cos^3(θ) ?

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

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prove cos(θ)-cos(θ)sin^2(θ)=cos^3(θ)

prove

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

True

Solution steps

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

Manipulating left side

cos (θ) - cos (θ) sin^{2} (θ)

Factor

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

cos (θ) - cos (θ) sin^{2} (θ)

Factor out common term

cos (θ)

= cos (θ) (1 - sin^{2} (θ))

= (1 - sin^{2} (θ)) cos (θ)

Rewrite using trig identities

(1 - sin^{2} (θ)) cos (θ)

Use the Pythagorean identity:

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

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

= cos^{2} (θ) cos (θ)

Simplify

cos^{2} (θ) cos (θ) : cos^{3} (θ)

cos^{2} (θ) cos (θ)

Apply exponent rule:

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

cos^{2} (θ) cos (θ) = cos^{2 + 1} (θ)

= cos^{2 + 1} (θ)

Add the numbers:

2 + 1 = 3

= cos^{3} (θ)

= cos^{3} (θ)

= cos^{3} (θ)

We showed that the two sides could take the same form

\Rightarrow True

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

csc (x) - 1 = \frac{cot ^{2} ( x )}{csc ( x ) + 1}

cot (x) tan (x) = (sec (x)) (csc (x))

csc^{2} (t) = cot^{2} (t) + 1

2 tan (x) = 1

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