(2sin10 cos10 cos 20) : (cos 50)

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To simplify this expression, we first apply the double angle identity for sine: sin(2θ) = 2sinθcosθ.

Using this identity, we rewrite the expression as:

2(sin10)(cos10)(cos20) = 2(sin10cos10)(cos20).

Next, we use the double angle identity for cosine: cos(2θ) = 2cos^2(θ) - 1.

Using this identity, we rewrite the expression again as:

2(1/2)(cos20 + cos40) = cos20 + cos40.

Finally, we use the sum-to-product identities to simplify cos20 + cos40. We have:

cos20 + cos40 = 2cos(30)cos(10) = 2(√3/2)(cos10) = √3cos10.

Therefore, the simplified expression is:

(2sin10cos10cos20) / cos50 = (√3cos10) / cos50 = √3(tan10).