Simplify cos(5x)

Problem

cos(5*x)

Solution

1. Apply the sum formula for cosine by splitting the argument into 4*x and x

cos(5*x)=cos(4*x+x)

cos(4*x+x)=cos(4*x)*cos(x)−sin(4*x)*sin(x)

2. Use double angle formulas to expand cos(4*x) and sin(4*x) in terms of 2*x

cos(4*x)=2*cos2(2*x)−1

sin(4*x)=2*sin(2*x)*cos(2*x)

3. Substitute these expansions back into the expression.

cos(5*x)=(2*cos2(2*x)−1)*cos(x)−(2*sin(2*x)*cos(2*x))*sin(x)

4. Apply double angle formulas again to reduce all terms to the argument x

cos(2*x)=2*cos2(x)−1

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

5. Substitute and expand the terms.

cos(5*x)=(2*(2*cos2(x)−1)2−1)*cos(x)−2*(2*sin(x)*cos(x))*(2*cos2(x)−1)*sin(x)

cos(5*x)=(2*(4*cos4(x)−4*cos2(x)+1)−1)*cos(x)−4*sin2(x)*cos(x)*(2*cos2(x)−1)

6. Use the identity sin2(x)=1−cos2(x) to ensure the expression is entirely in terms of cos(x)

cos(5*x)=(8*cos4(x)−8*cos2(x)+1)*cos(x)−4*(1−cos2(x))*cos(x)*(2*cos2(x)−1)

7. Distribute and combine like terms to reach the final simplified form.

cos(5*x)=8*cos5(x)−8*cos3(x)+cos(x)−(4*cos(x)−4*cos3(x))*(2*cos2(x)−1)

cos(5*x)=8*cos5(x)−8*cos3(x)+cos(x)−(8*cos3(x)−4*cos(x)−8*cos5(x)+4*cos3(x))

cos(5*x)=16*cos5(x)−20*cos3(x)+5*cos(x)

Final Answer

cos(5*x)=16*cos5(x)−20*cos3(x)+5*cos(x)

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