Expand the Trigonometric Expression cos(3x)

Problem

cos(3*x)

Solution

1. Apply the angle sum identity cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B) by rewriting 3*x as 2*x+x

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

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

2. Substitute the double angle identities cos(2*x)=2*cos2(x)−1 and sin(2*x)=2*sin(x)*cos(x) into the expression.

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

3. Distribute the terms and simplify the trigonometric products.

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

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

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

5. Expand the remaining parentheses and combine like terms.

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

cos(3*x)=4*cos3(x)−3*cos(x)

Final Answer

cos(3*x)=4*cos3(x)−3*cos(x)

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