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

cot^{4} (x) + cot^{2} (x) = cot^{2} (x) csc^{2} (x)

cot^{3} (x) = cot (x) (csc^{2} (x) - 1)

sin (2 a) = \frac{2 tan ( a )}{1 + tan ^{2} ( a )}

10 sec^{2} (θ) - 4 tan^{2} (θ) = 4 + 6 sec^{2} (θ)

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

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