Simplify

Problem

cos(π/16)*cos((3*π)/16)−sin(π/16)*sin((3*π)/16)

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 sum, which states cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

3. Substitute the values A=π/16 and B=(3*π)/16 into the sum formula.

4. Add the fractions inside the cosine function.

π/16+(3*π)/16=(4*π)/16

5. Simplify the resulting fraction.

(4*π)/16=π/4

6. Evaluate the cosine of the simplified angle using known trigonometric values.

cos(π/4)=√(,2)/2

Final Answer

cos(π/16)*cos((3*π)/16)−sin(π/16)*sin((3*π)/16)=√(,2)/2

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