Is cos(3α)=4cos^3(α)-3cos(α) ?

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

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prove cos(3α)=4cos^3(α)-3cos(α)

prove

cos (3 α) = 4 cos^{3} (α) - 3 cos (α)

True

Solution steps

cos (3 α) = 4 cos^{3} (α) - 3 cos (α)

Manipulating left side

cos (3 α)

Rewrite using trig identities

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

Use the Double Angle identity:

cos (2 α) = 2 cos^{2} (α) - 1

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

Use the Pythagorean identity:

cos^{2} (α) + sin^{2} (α) = 1

sin^{2} (α) = 1 - cos^{2} (α)

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

Expand

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

= 4 cos^{3} (α) - 3 cos (α)

We showed that the two sides could take the same form

\Rightarrow True

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

sin (x) cot (x) + cos (x) tan^{2} (x) = sec (x)

(cot^{2} (x) + 1) (1 - cos^{2} (x)) = 1

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

tan (2 π + θ) = tan (θ)

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