Is cos^3(θ)+cos(θ)sin^2(θ)=cos(θ) ?

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

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prove cos^3(θ)+cos(θ)sin^2(θ)=cos(θ)

prove

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

True

Solution steps

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

Manipulating left side

cos^{3} (θ) + cos (θ) sin^{2} (θ)

Factor

cos^{3} (θ) + cos (θ) sin^{2} (θ) : cos (θ) (cos^{2} (θ) + sin^{2} (θ))

cos^{3} (θ) + cos (θ) sin^{2} (θ)

Apply exponent rule:

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

cos^{3} (θ) = cos (θ) cos^{2} (θ)

= cos (θ) cos^{2} (θ) + cos (θ) sin^{2} (θ)

Factor out common term

cos (θ)

= cos (θ) (cos^{2} (θ) + sin^{2} (θ))

= (cos^{2} (θ) + sin^{2} (θ)) cos (θ)

Rewrite using trig identities

(cos^{2} (θ) + sin^{2} (θ)) cos (θ)

Use the Pythagorean identity:

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

= 1 \cdot cos (θ)

Simplify

= cos (θ)

= cos (θ)

We showed that the two sides could take the same form

\Rightarrow True

csc (θ) sin (θ) = 1

\frac{cos ( x ) - cos ( 9 x )}{sin ( x ) + sin ( 9 x )} = tan (4 x)

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

(1 - cos (x))^{2} = csc^{2} (x) - 2 cot (x)

cos^{2} (9 x) - sin^{2} (9 x) = cos (18 x)

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