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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

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

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

Factor out common term

sin (x)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Apply exponent rule:

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

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

= sin^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= sin^{3} (x)

= sin^{3} (x)

= sin^{3} (x)

We showed that the two sides could take the same form

\Rightarrow True

sec (x) = \frac{1}{cos ( x )}

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

tan (x) + sec (x) = \frac{cos ( x )}{1 - sin ( x )}

csc^{2} (x) - cot^{2} (x) = 1

cot (x) - tan (x) = 2 cot (2 x)

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