Sin50sin20+cos20sin40/cos40cos70+sin70cos50

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To simplify this expression, we can first rewrite the trigonometric functions using their double angle or sum/difference formulas.

sin50sin20 = 1/2[cos(50-20) - cos(50+20)]
= 1/2[cos30 - cos70]
= 1/2[(√3/2) - (cos70)]

cos20sin40 = 1/2[sin(20+40) + sin(40-20)]
= 1/2[sin60 + sin20]
= 1/2[(√3/2) + (sin20)]

cos40cos70 = 1/2[cos(40+70) + cos(40-70)]
= 1/2[cos110 + cos30]
= 1/2[(-sin20) + (√3/2)]

sin70cos50 = 1/2[sin(70+50) + sin(70-50)]
= 1/2[sin120 + sin20]
= 1/2[(√3/2) + (sin20)]

Now substituting these simplified expressions back into the original expression, we get:

(1/2[(√3/2) - cos70] + 1/2[(√3/2) + sin20]) / (1/2[(-sin20) + (√3/2)] + 1/2[(√3/2) + (sin20)])

Now we can simplify this expression further by combining like terms and using trigonometric identities.

(√3/2 - cos70 + √3/2 + sin20) / (-sin20 + √3/2 + √3/2 + sin20)
(2√3/2 + sin20 + sin20) / (sin20 - √3/2 + √3/2 + sin20)
(2√3 + 2sin20) / 2sin20
√3 + sin20 / sin20

Therefore, the simplified form of the expression Sin50sin20+cos20sin40/cos40cos70+sin70cos50 is √3 + cot20.