Use identities to identify the exact value of cos (-105)?

We can use the identity cos(-x) = cos(x) to find the exact value of cos(-105):

cos(-105) = cos(105)

Now, we can use the angle addition formula for cosine:

cos(105) = cos(60 + 45)

= cos(60)cos(45) - sin(60)sin(45)

= (1/2)(√2/2) - (√3/2)(√2/2)

= √2/4 - √6/4

= (√2 - √6)/4

Therefore, cos(-105) = (√2 - √6)/4.

Step 1: Apply the Cosine of a Negative Angle Identity

cos(-x) = cos(x)

Step 2: Replace x with -105

cos(-105) = cos(105)

Step 3: Apply the Cosine of a Supplementary Angle Identity

cos(180 - x) = -cos(x)

Step 4: Replace x with 105

cos(180 - 105) = -cos(105)

Step 5: Simplify

cos(75) = -cos(105)

Therefore, cos(-105) = -cos(105) = -0.2588.