Is cos^5(x)=(1-sin^2(x))^2cos(x) ?

The answer to whether cos^5(x)=(1-sin^2(x))^2cos(x) is True

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prove cos^5(x)=(1-sin^2(x))^2cos(x)

prove

cos^{5} (x) = (1 - sin^{2} (x))^{2} cos (x)

True

Solution steps

cos^{5} (x) = (1 - sin^{2} (x))^{2} cos (x)

Manipulating right side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

(cos^{2} (x))^{2} cos (x) : cos^{5} (x)

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

(cos^{2} (x))^{2} = cos^{4} (x)

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

Multiply the numbers:

2 \cdot 2 = 4

= cos^{4} (x)

= cos^{4} (x) cos (x)

Apply exponent rule:

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

cos^{4} (x) cos (x) = cos^{4 + 1} (x)

= cos^{4 + 1} (x)

Add the numbers:

4 + 1 = 5

= cos^{5} (x)

= cos^{5} (x)

= cos^{5} (x)

We showed that the two sides could take the same form

\Rightarrow True

Is cos^5(x)=(1-sin^2(x))^2cos(x) ?
The answer to whether cos^5(x)=(1-sin^2(x))^2cos(x) is True