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

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

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

prove

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

True

Solution steps

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

Manipulating right side

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

Factor

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

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

Apply exponent rule:

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

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

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

Factor out common term

sin^{3} (z)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

sec (x) = sin (x) \cdot (tan (x) + cot (x))

(cot (θ) + tan (θ)) sin (θ) = sec (θ)

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

cos (A) = 2 cos (A) sin (B)

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