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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = sin (x), b = 1, c = sin^{2} (x)

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

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

Simplify

1 \cdot sin (x) - sin^{2} (x) sin (x) : sin (x) - sin^{3} (x)

1 \cdot sin (x) - sin^{2} (x) sin (x)

1 \cdot sin (x) = sin (x)

1 \cdot sin (x)

Multiply:

1 \cdot sin (x) = sin (x)

= sin (x)

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

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

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

Add similar elements:

sin (x) - sin (x) = 0

= - sin^{3} (x)

= - sin^{3} (x)

= - sin^{3} (x)

We showed that the two sides could take the same form

\Rightarrow True

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