ЯсноПонятно24

To simplify the expression, we can expand the left side using the distributive property:

(sinx - cosx) (sqrt(3)siny + cosy) = sinx(sqrt(3)siny) - sinx(cosy) - cosx(sqrt(3)siny) + cosx(cosy) = sqrt(3)sinx^2 - sinxcosy - sqrt(3)cosxsinx + cosx^2

Since sin^2(x) + cos^2(x) = 1, we can replace sin^2(x) with 1 - cos^2(x) in the expression:

sqrt(3)(1 - cos^2(x)) - sinxcosy - sqrt(3)cosxsinx + cos^2(x) = sqrt(3) - sqrt(3)cos^2(x) - sinxcosy - sqrt(3)cosxsinx + cos^2(x)

Now, we can simplify further by using the trigonometric identity sin(2x) = 2sin(x)cos(x):

sqrt(3) - sqrt(3)cos^2(x) - sinxcosy - sqrt(3)cosxsinx + cos^2(x) = sqrt(3) - sqrt(3)cos^2(x) - 0.5sin(2x) - 0.5sin(2x) + cos^2(x) = sqrt(3) - sqrt(3)cos^2(x) - sin(2x) + cos^2(x)

Now, we can use the trigonometric identity cos(2x) = cos^2(x) - sin^2(x) to simplify further:

sqrt(3) - sqrt(3)cos^2(x) - sin(2x) + cos^2(x) = sqrt(3) - sqrt(3)cos^2(x) - sin(2x) + cos^2(x) = sqrt(3) - sqrt(3)cos^2(x) - sin(2x) + cos^2(x) = sqrt(3) - sqrt(3)cos^2(x) - sin(2x) + cos^2(x) = sqrt(3) - sqrt(3)cos^2(x) - sin(2x) + cos^2(x)

Therefore, the simplified expression is sqrt(3) - sqrt(3)cos^2(x) - sin(2x) + cos^2(x).