Simplify

Problem

cos((11*π)/12)*cos((2*π)/3)+sin((11*π)/12)*sin((2*π)/3)

Solution

1. Identify the trigonometric identity that matches the form of the expression. The expression follows the cosine difference identity:

cos(A)*cos(B)+sin(A)*sin(B)=cos(A−B)

2. Assign the values for the variables A and B from the given expression:

A=(11*π)/12

B=(2*π)/3

3. Substitute these values into the identity to condense the expression:

cos((11*π)/12−(2*π)/3)

4. Find a common denominator to subtract the fractions inside the cosine function:

(2*π)/3=(8*π)/12

5. Subtract the angles:

(11*π)/12−(8*π)/12=(3*π)/12

6. Simplify the resulting fraction:

(3*π)/12=π/4

7. Evaluate the cosine of the simplified angle using the unit circle:

cos(π/4)=√(,2)/2

Final Answer

cos((11*π)/12)*cos((2*π)/3)+sin((11*π)/12)*sin((2*π)/3)=√(,2)/2

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