Simplify cos(pi/7)cos(pi/5)-sin(pi/7)sin(pi/5)

Problem

cos(π/7)*cos(π/5)−sin(π/7)*sin(π/5)

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. Recall the cosine addition formula, which states:

cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

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

A=π/7

B=π/5

4. Apply the formula by substituting A and B into the cosine sum identity:

cos(π/7+π/5)

5. Find a common denominator for the fractions inside the cosine function. The least common multiple of 7 and 5 is 35

π/7=(5*π)/35

π/5=(7*π)/35

6. Add the fractions:

(5*π)/35+(7*π)/35=(12*π)/35

Final Answer

cos(π/7)*cos(π/5)−sin(π/7)*sin(π/5)=cos((12*π)/35)

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