Final answer:
To find the exact value of cos(-7π/12), we use the cosine sum identity, cos(a + b) = cos(a)cos(b) - sin(a)sin(b), with the known angles -π/3 and -π/4. After evaluating, we determine the correct answer is B: cos(-7π/12) = (√6 - √2) / 4.
Step-by-step explanation:
The exact value of cos(-7π/12) can be found by using sum or difference identities since -7π/12 is not a standard angle on the unit circle. We can express -7π/12 as the sum of -4π/12 (which is -π/3 and a known angle) and -3π/12 (which is -π/4, also a known angle). Using the cosine sum identity, cos(a + b) = cos(a)cos(b) - sin(a)sin(b), we get:
cos(-7π/12) = cos(-π/3 - π/4) = cos(-π/3)cos(-π/4) - sin(-π/3)sin(-π/4).
Since cosine is an even function, cos(-π/3) = cos(π/3) and cos(-π/4) = cos(π/4). Since sine is odd, sin(-π/3) = -sin(π/3), and sin(-π/4) = -sin(π/4). Plugging in the known values, we get:
cos(-7π/12) = (√3/2)(√2/2) - (-√3/2)(-1/√2) = (√3√2/4) + (3/2√2/2) = √6/4 + √2/4 = (√6 + √2)/4.
Therefore, the correct option is B: cos(-7π/12) = (√6 - √2) / 4.