Sin 37º.cos8° + cos 37° sin 8°sin 45° . cos15° - sin15º. cos 45º

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Let's simplify the expression step by step:

sin 37º cos 8°:
Using the trigonometric identity sin(a)cos(b) = (1/2)sin(a+b) + (1/2)sin(a-b):
sin 37º cos 8° = (1/2) sin (45º) + (1/2) * sin (29º)

cos 37° sin 8°:
Using the trigonometric identity cos(a)sin(b) = (1/2)[sin(a+b) - sin(a-b)]:
cos 37° sin 8° = (1/2) * [sin(45º) - sin(29º)]

sin 45° cos 15°:
sin 45° = 1/√2 and cos 15° = cos(45° - 30°) = cos45° cos30° + sin45° sin30°
Substituting these values, we get:
sin 45° cos 15° = (1/√2) [(√2/2)(√3/2) + (1/√2) * (1/2)] = 1/4 + √6/4

sin 15° cos 45°:
sin 15° = sin(45°-30°) = sin45° cos30° - cos45° sin30°
Substituting these values, we get:
sin 15° cos 45° = (1/√2) * (√3/2 - √2/2) = (√3 - √2)/4

Now, putting it all together:
[(1/2) sin 45º + (1/2) sin 29º] - (1/2) [sin 29º - sin 45º] + (1/4 + √6/4) - [√3 - √2)/4]
= (1/2) + (1/2) sin 29º + (1/4 + √6/4) - ((√3 - √2)/4)
= 1 + (1/2) * sin 29º + 1/4 + √6/4 - √3/4 + √2/4

So, the final result of the expression is:
1 + (1/2)*sin29° + 1/4 + √6/4 - √3/4 + √2/4