Simplify cos(pi/12)+cos((7pi)/12)

Problem

cos(π/12)+cos((7*π)/12)

Solution

1. Identify the sum-to-product formula for cosine, which states cos(A)+cos(B)=2*cos((A+B)/2)*cos((A−B)/2)

2. Assign the values A=π/12 and B=(7*π)/12 to the formula.

3. Calculate the sum and difference of the angles.

(A+B)/2=(π/12+(7*π)/12)/2=(8*π)/12/2=(2*π)/3/2=π/3

(A−B)/2=(π/12−(7*π)/12)/2=(−(6*π)/12)/2=(−π/2)/2=−π/4

4. Substitute these values back into the sum-to-product formula.

cos(π/12)+cos((7*π)/12)=2*cos(π/3)*cos(−π/4)

5. Apply the even function property cos(−x)=cos(x) and evaluate the trigonometric constants.

cos(π/3)=1/2

cos(−π/4)=cos(π/4)=√(,2)/2

6. Multiply the results to find the final simplified value.

2⋅1/2⋅√(,2)/2=√(,2)/2

Final Answer

cos(π/12)+cos((7*π)/12)=√(,2)/2

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