Sin225cos290tg165 / ctg105cos60sin340

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To simplify this expression, we need to first convert all the trigonometric functions into sine and cosine.

Given:
sin225 = -sin45
cos290 = cos70
tg165 = -cot75
ctg105 = cot75
cos60 = 1/2
sin340 = -sin20

Therefore, the expression becomes:
-sin45 cos70 -cot75 / cot75 (1/2) -sin20

Now, simplify the expression:
= sin45 cos70 cot75 (1/2) sin20
= (√2/2) (cos70/sin70) (cos75/sin75) (1/2) (-√2/2)
= (√2 / 2) (cos70 / sin70) (cos75 / sin75) (-√2 / 4)
= (√2 / 2) (sin20) (cos75 / sin75) (-√2 / 4)
= (√2 / 2) (sin20) ((√3 / 2) / (1/2)) (-√2 / 4)
= (√2 / 2) (sin20) (√3) (-√2 / 4)
= - (√6) * (sin20) / 4

Therefore, the simplified expression is - (√6) * (sin20) / 4.