Is sin(3x)=3sin(x)-4(sin(x))^3 ?

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

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prove sin(3x)=3sin(x)-4(sin(x))^3

prove

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

True

Solution steps

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

Manipulating left side

sin (3 x)

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

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

Expand

(1 - 2 sin^{2} (x)) sin (x) + 2 (1 - sin^{2} (x)) sin (x) : - 4 sin^{3} (x) + 3 sin (x)

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

We showed that the two sides could take the same form

\Rightarrow True

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

cot (x) \cdot sec (x) = csc (x)

2 - csc^{2} (θ) = 1 - cot^{2} (θ)

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

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

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