Is sin^9(x)=(1-cos^2(x))^4sin(x) ?

The answer to whether sin^9(x)=(1-cos^2(x))^4sin(x) is True

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prove sin^9(x)=(1-cos^2(x))^4sin(x)

prove

sin^{9} (x) = (1 - cos^{2} (x))^{4} sin (x)

True

Solution steps

sin^{9} (x) = (1 - cos^{2} (x))^{4} sin (x)

Manipulating right side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

(sin^{2} (x))^{4} sin (x) : sin^{9} (x)

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

(sin^{2} (x))^{4} = sin^{8} (x)

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

Apply exponent rule:

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

= sin^{2 \cdot 4} (x)

Multiply the numbers:

2 \cdot 4 = 8

= sin^{8} (x)

= sin^{8} (x) sin (x)

Apply exponent rule:

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

sin^{8} (x) sin (x) = sin^{8 + 1} (x)

= sin^{8 + 1} (x)

Add the numbers:

8 + 1 = 9

= sin^{9} (x)

= sin^{9} (x)

= sin^{9} (x)

We showed that the two sides could take the same form

\Rightarrow True

Is sin^9(x)=(1-cos^2(x))^4sin(x) ?
The answer to whether sin^9(x)=(1-cos^2(x))^4sin(x) is True