1 Answers
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.
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