Final answer:
The exact value of cosine (105 degrees) is (√6 - √2)/4.
Step-by-step explanation:
To find the exact value of cosine (105 degrees), we can use the identity cos(180 - x) = -cos(x). So, cos(105 degrees) = -cos(75 degrees). We can further use the identity cos(180 - x) = cos(x) to simplify it to cos(105 degrees) = cos(75 degrees).
Now, we know that cos(30 degrees) = √3/2 and cos(45 degrees) = √2/2. Using the sum and difference of angles identities, we can find that cos(75 degrees) = cos(45 degrees + 30 degrees) = cos(45 degrees)cos(30 degrees) - sin(45 degrees)sin(30 degrees) = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4.
Therefore, the exact value of cosine (105 degrees) is (√6 - √2)/4.