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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sin (3 x)

Rewrite using trig identities

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

Expand

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

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{cot ( x ) + tan ( x )}{sec ( x )} = csc (x)

\frac{cos ( 3 x ) - cos ( 5 x )}{sin ( 3 x ) + sin ( 5 x )} = tan (x)

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

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

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

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