The simplified exact values for the given trigonometric expressions are sin 75° cos 105° + cos 75° sin 105° for part (a) and sqrt(3)/3 + sqrt(2) for part (b).
a) For the expression (sin 45°) (cos 45°) + (sin 30°) (cos 60°):
Using the angle sum identity for sine, sin(A + B) = sin A cos B + cos A sin B, we can rewrite the expression as follows:
(sin 45° + 30°) (cos 45° + 60°)
Now, applying the angle sum identities for sine and cosine, we get:
sin (45° + 30°) = sin 75°
cos (45° + 60°) = cos 105°
So, the expression becomes:
sin 75° cos 105° + cos 75° sin 105°
These angles do not have standard reference angles, so the expressions cannot be simplified further without a calculator.
b) For the expression (1 - tan 45°) (sin 30°) (cos 30°) (tan 60°) + tan 30° + 2(sin 45°) (cos 60°):
First, simplify each term:
(1 - 1) (1/2) (sqrt(3)/2) (sqrt(3)) + sqrt(3)/3 + 2(sqrt(2)/2) (1/2)
This simplifies to:
0 + sqrt(3)/3 + sqrt(2)
So, the exact value of the given expression is sqrt(3)/3 + sqrt(2).