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

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

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

prove

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

True

Solution steps

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

Manipulating right 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

cos (2 x) - 3 sin^{2} (x) = 1 - 5 sin^{2} (x)

\frac{1 - sin ( x )}{1 - \frac{1}{s i n ( x )}} = - sin (x)

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

csc (x) - sin (x) = \frac{cot ^{2} ( x )}{csc ( x )}

\frac{cot ( 2 θ )}{cos ( 2 θ )} = csc (2 θ)

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