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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Add similar elements:

sin^{3} (x) - sin^{3} (x) = 0

= sin (x)

= sin (x)

= sin (x)

We showed that the two sides could take the same form

\Rightarrow True

\frac{2}{cot ( x ) + tan ( x )} = sin (2 x)

tan (t) + 2 cos (t) csc (t) = sec (t) csc (t) + cot (t)

\frac{1}{tan ( x ) ( 1 + cos ( 2 x ) )} = csc (2 x)

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

\frac{tan ^{2} ( x )}{sec ( x ) + 1} = sec (x) - 1

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