Simplify

Problem

sin((5*π)/12)*cos(π/4)−cos((5*π)/12)*sin(π/4)

Solution

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

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

3. Substitute the values A=(5*π)/12 and B=π/4 into the identity.

sin((5*π)/12−π/4)

4. Find a common denominator to subtract the fractions inside the argument.

π/4=(3*π)/12

5. Subtract the angles.

(5*π)/12−(3*π)/12=(2*π)/12

6. Simplify the resulting fraction.

(2*π)/12=π/6

7. Evaluate the sine of the simplified angle using the unit circle or standard values.

sin(π/6)=1/2

Final Answer

sin((5*π)/12)*cos(π/4)−cos((5*π)/12)*sin(π/4)=1/2

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