Evaluate giving exact values and identities. Calculator based solutions are not acceptable. Thanks for any help

cos (-5pi/12)

5pi/12 radians = 75º

so cos(-5pi/12) = cos(-75º = cos 75º
= cos(45+30)º
= cos45cos30 - sin45sin30
= √2/2*√3/2 - √/2*1/2
= (√6 - √2)/4

To evaluate cos (-5π/12) without relying on a calculator, we can make use of the trigonometric identities and special angles.

The given angle, -5π/12, falls in the second quadrant. To evaluate cos (-5π/12), we can rewrite the angle as the sum of two special angles: -π/3 and -π/4. This is because -π/3 + (-π/4) = -5π/12.

Now, let's consider the cosine of -π/3 and -π/4:

cos(-π/3) = cos(-60°) = 1/2

cos(-π/4) = cos(-45°) = √2/2

Using the sum formula for cosine, we have:

cos(-5π/12) = cos(-π/3 - π/4) = cos(-π/3)cos(-π/4) + sin(-π/3)sin(-π/4)

Since we've already found the values for cos(-π/3) and cos(-π/4), we can substitute those in:

cos(-5π/12) = (1/2)(√2/2) + (√3/2)(√2/2)

To simplify further, we can rationalize the denominator by multiplying both the numerator and denominator of (√2/2) by √2:

cos(-5π/12) = (1/2)(√2/2) + (√3/2)(√2/2)
= √2/4 + (√6)/4
= (√2 + √6)/4

Therefore, the exact value of cos(-5π/12) is (√2 + √6)/4.