Final Answer:
The exact value of cos(5π/12) is (√6 + √2) / 4.
Step-by-step explanation:
To find the exact value of cos(5π/12) using the sum and difference of cosines formula, we can express 5π/12 as the sum of two angles, say A and B, such that A + B = 5π/12. A suitable choice is A = π/4 and B = π/3, as their sum indeed equals 5π/12.
Now, the sum and difference of cosines formula is given by:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
Substituting the values, we get:
cos(5π/12) = cos(π/4 + π/3) = cos(π/4)cos(π/3) - sin(π/4)sin(π/3).
Since cos(π/4) = √2/2, cos(π/3) = 1/2, sin(π/4) = √2/2, and sin(π/3) = √3/2, we can substitute these values into the formula:
cos(5π/12) = (√2/2 * 1/2) - (√2/2 * √3/2).
Simplifying further:
cos(5π/12) = (√2 + √6) / 4.
Thus, the final exact value of cos(5π/12) is (√6 + √2) / 4, which can be obtained by applying the sum and difference of cosines formula and simplifying the expression.