Find the exact value for cos π 12 applying sum and difference formulas involving π 3 and π 4.

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, π/12 can be split into π/3-π/4.

cos (π/3-π/4)

Use the difference formula for cosine to simplify the expression. The formula states that cos (A-B) = cos (A) cos (B) + sin (A) sin (B)

cos (π/3) ⋅cos (π/4) + sin (π/3) ⋅sin (π/4)

The exact value of cos (π/3) is 12, so:

(12) ⋅cos (π/4) + sin (π/3) ⋅sin (π/4)

The exact value of cos (π/4) is √22.

(12) ⋅ (√22) + sin (π/3) ⋅sin (π/4)

The exact value of sin (π/3) is √32.

(12) ⋅ (√22) + (√32) ⋅sin (π/4)

The exact value of sin (π/4) is √22.

(12) ⋅ (√22) + (√32) ⋅ (√22)

Simplify each term:

√24+√64

Combine the numerators over the common denominator.

(√2+√6) / 4