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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sin (3 a)

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

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

Expand

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

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{sin ( x ) + tan ( x )}{1 + cos ( x )} = tan (x)

\frac{1}{tan ( θ ) csc ( θ )} = cos (θ)

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

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

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

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