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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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 right side

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

Factor

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

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

Factor out common term

cos (x)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Apply exponent rule:

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

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

= cos^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= cos^{3} (x)

= cos^{3} (x)

= cos^{3} (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