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Abdirahman Omar

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

Like1 Isticmaale
1 Isticmaale

1 Answer

  1. 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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