Is cos^3(x)+cos(x)sin^2(x)=cos(x) ?

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

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prove cos^3(x)+cos(x)sin^2(x)=cos(x)

prove

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

True

Solution steps

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

Manipulating left side

cos^{3} (x) + cos (x) sin^{2} (x)

Factor

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

cos^{3} (x) + cos (x) sin^{2} (x)

Apply exponent rule:

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

cos^{3} (x) = cos (x) cos^{2} (x)

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

Factor out common term

cos (x)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= 1 \cdot cos (x)

Simplify

= cos (x)

= cos (x)

We showed that the two sides could take the same form

\Rightarrow True

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