To solve this trigonometric expression, we can use the product-to-sum identities and the double angle identities.

Given expression: sin(45° A) cos(45° B) + cos(45° A) sin(45° B)

Using the product-to-sum identity, sin(A) cos(B) = (1/2)[sin(A + B) + sin(A - B)], we can rewrite the expression as:

(1/2)[sin(45° A + 45° B) + sin(45° A - 45° B)] + (1/2)[sin(45° A + 45° B) - sin(45° A - 45° B)]

Now, let's simplify each term separately:

1) (1/2)[sin(45° A + 45° B) + sin(45° A - 45° B)]
Using the double angle identity, sin(α + β) = sin(α)cos(β) + cos(α)sin(β), we can rewrite the first term as:
sin(45° A + 45° B) = sin(45°)(cos(45° A)cos(45° B) + sin(45° A)sin(45° B))

Substituting this back into the expression, we have:
(1/2)[sin(45°)(cos(45° A)cos(45° B) + sin(45° A)sin(45° B)) + sin(45° A - 45° B)]

Now, let's consider the second term:

2) (1/2)[sin(45° A + 45° B) - sin(45° A - 45° B)]
Using the double angle identity, sin(α - β) = sin(α)cos(β) - cos(α)sin(β), we can rewrite the second term as:
sin(45° A - 45° B) = sin(45°)(cos(45° A)cos(45° B) - sin(45° A)sin(45° B))

Substituting this back into the expression, we have:
(1/2)[sin(45°)(cos(45° A)cos(45° B) + sin(45° A)sin(45° B)) - sin(45°)(cos(45° A)cos(45° B) - sin(45° A)sin(45° B))]

Now, let's simplify the expression:

(1/2)[sin(45°)(cos(45° A)cos(45° B) + sin(45° A)sin(45° B)) + sin(45°)(cos(45° A)cos(45° B) - sin(45° A)sin(45° B))]

Removing the common factor of sin(45°), we have:
(1/2)[sin(45°)(2cos(45° A)cos(45° B))]

Since sin(45°) = cos(45°), we can simplify further:
(1/2)(2cos(45° A)cos(45° B))

Finally, simplifying further:
cos(45° A)cos(45° B)

Therefore, the correct answer is option B: cos(A B).

There's a formula sinAcosB + cosAsinB = sin(A+B) so, here in this question, assume (45-A) as A and (45-B) as B so you'll get sin( 90 -(A+B)) which is cos (A+B)