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

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

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

prove

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

True

Solution steps

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

Manipulating right side

cos (3 a)

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

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

Expand

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

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

We showed that the two sides could take the same form

\Rightarrow True

sin^{2} (θ) + (\frac{1}{2})^{2} = 1

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

sin (θ) (cot (θ) + tan (θ)) = tan (θ)

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

\frac{sec ( x ) cos ( x )}{cot ( x )} = tan (x)

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