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

sin (a) cot (a) = cos (a)

cos (3 θ) = cos^{3} (θ) - 3 sin^{2} (θ) cos (θ)

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

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

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

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