Verify the Identity cos(pi/3+x)+cos(pi/3-x)=cos(x)

Problem

cos(π/3+x)+cos(π/3−x)=cos(x)

Solution

1. Apply the sum and difference formulas for cosine, which are cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B) and cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B)

2. Expand the left side of the equation using these identities where A=π/3 and B=x

cos(π/3)*cos(x)−sin(π/3)*sin(x)+cos(π/3)*cos(x)+sin(π/3)*sin(x)

3. Combine like terms by noticing that the sine terms cancel each other out.

2*cos(π/3)*cos(x)

4. Substitute the exact value for cos(π/3) which is 1/2

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

5. Simplify the expression to show that the left side equals the right side.

cos(x)

Final Answer

cos(π/3+x)+cos(π/3−x)=cos(x)

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