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

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

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

prove

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

True

Solution steps

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

Manipulating left 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

2 cos (x) sin (y) = sin (x + y) - sin (x - y)

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

\frac{tan ( 2 x ) + cot ( 2 x )}{csc ( 2 x )} = sec (2 x)

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

1 - 4 sin^{2} (x) cos^{2} (x) = cos^{2} (2 x)

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