Is sin^7(x)=(1-cos^2(x))^3sin(x) ?

The answer to whether sin^7(x)=(1-cos^2(x))^3sin(x) is True

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prove sin^7(x)=(1-cos^2(x))^3sin(x)

prove

sin^{7} (x) = (1 - cos^{2} (x))^{3} sin (x)

True

Solution steps

sin^{7} (x) = (1 - cos^{2} (x))^{3} sin (x)

Manipulating right side

(1 - cos^{2} (x))^{3} sin (x)

Rewrite using trig identities

(1 - cos^{2} (x))^{3} sin (x)

Use the Pythagorean identity:

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

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

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

Simplify

(sin^{2} (x))^{3} sin (x) : sin^{7} (x)

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

(sin^{2} (x))^{3} = sin^{6} (x)

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

Apply exponent rule:

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

= sin^{2 \cdot 3} (x)

Multiply the numbers:

2 \cdot 3 = 6

= sin^{6} (x)

= sin^{6} (x) sin (x)

Apply exponent rule:

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

sin^{6} (x) sin (x) = sin^{6 + 1} (x)

= sin^{6 + 1} (x)

Add the numbers:

6 + 1 = 7

= sin^{7} (x)

= sin^{7} (x)

= sin^{7} (x)

We showed that the two sides could take the same form

\Rightarrow True

tan (A) + cot (A) = sec (A) csc (A)

\frac{sec ( θ )}{csc ( θ )} = tan (θ)

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

tan (θ) \cdot cos (θ) = sin (θ)

csc (x) + cot (x) = \frac{1 + cos ( x )}{sin ( x )}

Is sin^7(x)=(1-cos^2(x))^3sin(x) ?
The answer to whether sin^7(x)=(1-cos^2(x))^3sin(x) is True