Simplify

Problem

cos((13*π)/15)*cos(−π/5)−sin((13*π)/15)*sin(−π/5)

Solution

1. Identify the trigonometric identity for the cosine of a sum, which states cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

2. Assign the values from the expression to the variables in the identity, where A=(13*π)/15 and B=−π/5

3. Apply the formula to rewrite the expression as a single cosine function.

cos((13*π)/15+(−π/5))

4. Find a common denominator to combine the terms inside the cosine function.

(13*π)/15−(3*π)/15

5. Subtract the fractions.

(10*π)/15

6. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 5.

(2*π)/3

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

cos((2*π)/3)=−1/2

Final Answer

cos((13*π)/15)*cos(−π/5)−sin((13*π)/15)*sin(−π/5)=−1/2

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