Is sin^5(x)=(1-2cos^2(x)+cos^4(x))(sin(x)) ?

The answer to whether sin^5(x)=(1-2cos^2(x)+cos^4(x))(sin(x)) is True

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prove sin^5(x)=(1-2cos^2(x)+cos^4(x))(sin(x))

prove

sin^{5} (x) = (1 - 2 cos^{2} (x) + cos^{4} (x)) (sin (x))

True

Solution steps

sin^{5} (x) = (1 - 2 cos^{2} (x) + cos^{4} (x)) sin (x)

Manipulating right side

(1 - 2 cos^{2} (x) + cos^{4} (x)) sin (x)

Rewrite using trig identities

(1 - 2 cos^{2} (x) + cos^{4} (x)) sin (x)

Apply Perfect Square Formula:

a^{2} - 2 ab + b^{2} = (a - b)^{2}

1 - 2 cos^{2} (x) + cos^{4} (x) = (1 - cos^{2} (x))^{2}

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

Use the Pythagorean identity:

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

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

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

Simplify

(sin^{2} (x))^{2} : sin^{4} (x)

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

Apply exponent rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= sin^{4} (x)

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

Simplify

sin^{4} (x) sin (x) : sin^{5} (x)

sin^{4} (x) sin (x)

Apply exponent rule:

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

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

= sin^{4 + 1} (x)

Add the numbers:

4 + 1 = 5

= sin^{5} (x)

= sin^{5} (x)

= sin^{5} (x)

We showed that the two sides could take the same form

\Rightarrow True

Is sin^5(x)=(1-2cos^2(x)+cos^4(x))(sin(x)) ?
The answer to whether sin^5(x)=(1-2cos^2(x)+cos^4(x))(sin(x)) is True