Is sin(3x)=sin(x)(4cos^2(x)-1) ?

The answer to whether sin(3x)=sin(x)(4cos^2(x)-1) is True

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prove sin(3x)=sin(x)(4cos^2(x)-1)

prove

sin (3 x) = sin (x) (4 cos^{2} (x) - 1)

True

Solution steps

sin (3 x) = sin (x) (4 cos^{2} (x) - 1)

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)

We showed that the two sides could take the same form

\Rightarrow True

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

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

cos (x) - csc (x) cot (x) = - cos (x) cot^{2} (x)

(cos (θ) - 1) (cos (θ) + 1) = - sin^{2} (θ)

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

Is sin(3x)=sin(x)(4cos^2(x)-1) ?
The answer to whether sin(3x)=sin(x)(4cos^2(x)-1) is True