Is sin(3x)=4sin(x)*cos^2(x)-sin(x) ?

The answer to whether sin(3x)=4sin(x)*cos^2(x)-sin(x) is True

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prove sin(3x)=4sin(x)*cos^2(x)-sin(x)

prove

sin (3 x) = 4 sin (x) \cdot cos^{2} (x) - sin (x)

True

Solution steps

sin (3 x) = 4 sin (x) cos^{2} (x) - sin (x)

Manipulating left side

sin (3 x)

Rewrite using trig identities

= 3 sin (x) - 4 sin^{3} (x)

Factor

3 sin (x) - 4 sin^{3} (x) : sin (x) (3 - 4 sin^{2} (x))

= (3 - 4 sin^{2} (x)) sin (x)

Rewrite using trig identities

= sin (x) (4 cos^{2} (x) - 1)

Expand

(- 1 + 4 cos^{2} (x)) sin (x) : - sin (x) + 4 cos^{2} (x) sin (x)

= - sin (x) + 4 cos^{2} (x) sin (x)

= 4 sin (x) cos^{2} (x) - sin (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (θ) = \frac{4}{3} sin^{3} (θ) + \frac{1}{3} sin^{3} (θ)

sin (x - \frac{3 π}{2}) + sin (\frac{π}{2} + x) = 2 cos (x)

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

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

sec^{2} (x) cot^{2} (x) = csc^{2} (x)

Is sin(3x)=4sin(x)*cos^2(x)-sin(x) ?
The answer to whether sin(3x)=4sin(x)*cos^2(x)-sin(x) is True