Simplify

Problem

cos((7*π)/12)*cos(π/4)+sin((7*π)/12)*sin(π/4)

Solution

1. Identify the trigonometric identity that matches the structure of the expression. The expression follows the form cos(A)*cos(B)+sin(A)*sin(B)

2. Apply the formula for the cosine of a difference, which states cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B)

3. Substitute the values A=(7*π)/12 and B=π/4 into the identity.

cos((7*π)/12−π/4)

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

π/4=(3*π)/12

5. Subtract the fractions.

(7*π)/12−(3*π)/12=(4*π)/12

6. Simplify the resulting fraction.

(4*π)/12=π/3

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

cos(π/3)=1/2

Final Answer

cos((7*π)/12)*cos(π/4)+sin((7*π)/12)*sin(π/4)=1/2

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