Simplify cos(2x)cos(x)

Problem

cos(2*x)*cos(x)

Solution

1. Apply the double angle identity for cos(2*x) We use the identity cos(2*x)=2*cos2(x)−1 to express the entire function in terms of cos(x)

cos(2*x)*cos(x)=(2*cos2(x)−1)*cos(x)

2. Distribute the cos(x) term into the parentheses.

2*cos2(x)⋅cos(x)−1⋅cos(x)

3. Simplify the expression by combining the powers of the cosine terms.

2*cos3(x)−cos(x)

4. Alternative approach using the product-to-sum identity cos(A)*cos(B)=1/2*(cos(A−B)+cos(A+B))

cos(2*x)*cos(x)=1/2*(cos(2*x−x)+cos(2*x+x))

5. Simplify the arguments inside the cosine functions.

1/2*(cos(x)+cos(3*x))

Final Answer

cos(2*x)*cos(x)=1/2*(cos(x)+cos(3*x))

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