Simplify cos(3a)

Problem

cos(3*a)

Solution

1. Apply the sum identity for cosine, cos(x+y)=cos(x)*cos(y)−sin(x)*sin(y) by splitting the angle into 2*a and a

cos(3*a)=cos(2*a+a)

cos(3*a)=cos(2*a)*cos(a)−sin(2*a)*sin(a)

2. Substitute double angle identities for cos(2*a) and sin(2*a) using cos(2*a)=2*cos2(a)−1 and sin(2*a)=2*sin(a)*cos(a)

cos(3*a)=(2*cos2(a)−1)*cos(a)−(2*sin(a)*cos(a))*sin(a)

3. Distribute and simplify the terms in the expression.

cos(3*a)=2*cos3(a)−cos(a)−2*sin2(a)*cos(a)

4. Use the Pythagorean identity sin2(a)=1−cos2(a) to express the entire equation in terms of cos(a)

cos(3*a)=2*cos3(a)−cos(a)−2*(1−cos2(a))*cos(a)

5. Expand and combine like terms to reach the final simplified form.

cos(3*a)=2*cos3(a)−cos(a)−2*cos(a)+2*cos3(a)

cos(3*a)=4*cos3(a)−3*cos(a)

Final Answer

cos(3*a)=4*cos3(a)−3*cos(a)

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