Is sin^3(x)cos^2(x)=sin^3(x)-sin^5(x) ?

The answer to whether sin^3(x)cos^2(x)=sin^3(x)-sin^5(x) is True

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prove sin^3(x)cos^2(x)=sin^3(x)-sin^5(x)

prove

sin^{3} (x) cos^{2} (x) = sin^{3} (x) - sin^{5} (x)

True

Solution steps

sin^{3} (x) cos^{2} (x) = sin^{3} (x) - sin^{5} (x)

Manipulating right side

sin^{3} (x) - sin^{5} (x)

Factor

sin^{3} (x) - sin^{5} (x) : sin^{3} (x) (1 - sin^{2} (x))

sin^{3} (x) - sin^{5} (x)

Apply exponent rule:

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

sin^{5} (x) = sin^{3} (x) sin^{2} (x)

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

Factor out common term

sin^{3} (x)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

sec (2 a) = \frac{csc ^{2} ( a )}{cot ^{2} ( a ) - 1}

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

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

sin (0) - 1 = - 1

Is sin^3(x)cos^2(x)=sin^3(x)-sin^5(x) ?
The answer to whether sin^3(x)cos^2(x)=sin^3(x)-sin^5(x) is True