Howdo(tanx-1)(sin2x-2cos²x)= 2(1-2sinxcosx)

We can start by using the trigonometric identity: sin(2x) = 2sin(x)cos(x) We can substitute this identity into the expression: (tan(x) - 1)(sin(2x) - 2cos²(x)) = (tan(x) - 1)(2sin(x)cos(x) - 2cos²(x)) (substituting sin(2x) with 2sin(x)cos(x)) = -2cos(x)(tan(x) - 1)(cos(x) - sin(x)) (factoring out a -2cos(x) from the second term) = -2cos(x)(sin(x) - cos(x))(tan(x) - 1)(-1) (multiplying the second term by -1 and rearranging) = 2cos(x)(cos(x) - sin(x))(1 - tan(x)) = 2cos(x)(cos(x) - sin(x))(1 - sin(x)/cos(x)) (substituting tan(x) with sin(x)/cos(x)) = 2(cos(x) - sin(x))(cos(x)/cos(x) - sin(x)/cos(x)) = 2(cos(x) - sin(x))(1 - sin(x)cos(x)/(cos(x))^2) = 2(cos(x) - sin(x))(1 - sin(x)cos(x)) = 2(cos(x) - sin(x) + sin(x)cos(x) - sin(x)²cos(x)) = 2(1 - sin(x)cos(x) - 2sin(x)cos(x)) = 2(1 - 3sin(x)cos(x)) = 2(1 - 2sin(x)cos(x) - sin(x)cos(x)) = 2(1 - 2sin(x)cos(x)) (since sin(x)cos(x) = 1/2 sin(2x)) Therefore, (tan(x) - 1)(sin(2x) - 2cos²(x)) = 2(1 - 2sin(x)cos(x)) as required.