Solution:

Given expression: 2(sin 1 sin 2 . sin 89)÷ 2 (cos 1 cos 2 . cos 44)

Let's simplify the expression step by step:

1. Simplifying the numerator:
- Using the product-to-sum identity, sin A sin B = (1/2)(cos(A - B) - cos(A + B))
- Applying the product-to-sum identity to the given expression:
2(sin 1 sin 2 . sin 89) = 2(sin 1 sin 2) sin 89
= 2(1/2)(cos(1 - 2) - cos(1 + 2)) sin 89
= cos(-1) - cos(3) (sin 89 = 1)
= cos(1) - cos(3) (cos(-x) = cos(x))
So, the numerator simplifies to: cos(1) - cos(3)

2. Simplifying the denominator:
- Using the product-to-sum identity, cos A cos B = (1/2)(cos(A - B) + cos(A + B))
- Applying the product-to-sum identity to the given expression:
2(cos 1 cos 2 . cos 44) = 2(cos 1 cos 2) cos 44
= 2(1/2)(cos(1 - 2) + cos(1 + 2)) cos 44
= cos(-1) + cos(3) (cos 44 = cos(45 - 1) = cos(46 - 2))
= cos(1) + cos(3) (cos(-x) = cos(x))
So, the denominator simplifies to: cos(1) + cos(3)

3. Simplifying the expression:
We can now substitute the simplified numerator and denominator back into the original expression:
2(sin 1 sin 2 . sin 89)÷ 2 (cos 1 cos 2 . cos 44) = (cos(1) - cos(3))/(cos(1) + cos(3))

4. Applying the cosine sum-to-product identity:
cos(A) - cos(B) = -2 sin((A + B)/2) sin((A - B)/2)
Using this identity, we can rewrite the numerator as:
cos(1) - cos(3) = -2 sin((1 + 3)/2) sin((1 - 3)/2)
= -2 sin(2) sin(-1)
= 2 sin(2) sin(1) (sin(-x) = -sin(x))

Similarly, the denominator can be rewritten using the same identity:
cos(1) + cos(3) = 2 cos((1 + 3)/2) cos((1 - 3)/2)
= 2 cos(2) cos(-1)
= 2 cos(2) cos(1) (cos(-x) = cos(x))

So, the expression becomes: (2 sin(2) sin(1))/(2 cos(2) cos(1))

5. Canceling out the common factors:
The 2 in the numerator and denominator cancels out, leaving