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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sin (3 v)

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

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

Expand

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

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

We showed that the two sides could take the same form

\Rightarrow True

2 sin (x + \frac{π}{4}) = sin (x) + cos (x)

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

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

\frac{sin ( x )}{3 sin ( 2 x )} = \frac{1}{2 cos ( x )}

sin (x) tan (x) = 2 sec (x)

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