SOLUTION: Prove cos 3(theta) = 4 cos^3 (theta)

SOLUTION: Prove cos 3(theta) = 4 cos^3 (theta) - 3cos (theta)

Question 1142052: Prove cos 3(theta) = 4 cos^3 (theta) - 3cos (theta)
Answer by ikleyn(52802) (Show Source): You can put this solution on YOUR website!
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This problem is to be solved in 2 easy steps.

Step 1

cos(2a) = cos(a + a) = cos(a)*cos(a) - sin(a)*sin(a) = cos^2(a) - sin^2(a) = cos^2(a) - (1-cos^2(a)) = 2*cos^2(a) - 1.

Step 2

cos(3a) = cos(2a + a) = cos(2a)*cos(a) - sin(2a)*sin(a) = (2cos^2(a)-1)*cos(a) - (2*sin(a)*cos(a))*sin(a) = 

                      = 2*cos^3(a) - cos(a) - 2*sin^2(a)*cos(a) = 2*cos^3(a) - cos(a) - 2*(1-cos^2(a))*cos(a) = 

                      = 2*cos^3(a) - cos(a) - 2*cos(a) + 2*cos^2(a) = 4*cos^3(a) - 3*cos(a).

It is what has to be proved.

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