Expand the Trigonometric Expression cos(4x)

Problem

cos(4*x)

Solution

1. Apply the double angle formula for cosine, which states cos(2*θ)=2*cos2(θ)−1 Let θ=2*x

cos(4*x)=2*cos2(2*x)−1

2. Substitute the double angle formula again for the cos(2*x) term inside the square.

cos(4*x)=2*(2*cos2(x)−1)2−1

3. Expand the squared binomial (2*cos2(x)−1)2 using the pattern (a−b)2=a2−2*a*b+b2

cos(4*x)=2*(4*cos4(x)−4*cos2(x)+1)−1

4. Distribute the constant 2 through the parentheses.

cos(4*x)=8*cos4(x)−8*cos2(x)+2−1

5. Simplify the expression by combining the constant terms.

cos(4*x)=8*cos4(x)−8*cos2(x)+1

Final Answer

cos(4*x)=8*cos4(x)−8*cos2(x)+1

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