Mathematics

Question

use the power-reducing formulas as many times as possible to rewrite the expression in terms of the first power of the cosine. sin^4 3x cos^2 3x

1 Answer

  • [tex]\sin^43x\cos^23x[/tex]

    Pythagorean identity:

    [tex](1-\cos^23x)^2\cos^23x[/tex]

    Expand:

    [tex](1-2\cos^23x+\cos^43x)\cos^23x[/tex]

    Distribute:

    [tex]\cos^23x-2\cos^43x+\cos^63x[/tex]

    Half-angle identity for cosine:

    [tex]\dfrac{1+\cos6x}2-2\left(\dfrac{1+\cos6x}2\right)^2+\left(\dfrac{1+\cos6x}2\right)^3[/tex]

    Expand:

    [tex]\dfrac12+\dfrac12\cos6x-\dfrac12\left(1+2\cos6x+\cos^26x\right)+\dfrac18\left(1+3\cos6x+3\cos^26x+\cos^36x\right)[/tex]

    Simplify:

    [tex]\dfrac18-\dfrac18\cos6x-\dfrac18\cos^26x+\dfrac18\cos^36x[/tex]

    Half-angle again:

    [tex]\dfrac18-\dfrac18\cos6x-\dfrac18\left(\dfrac{1+\cos12x}2\right)+\dfrac18\cos6x\left(\dfrac{1+\cos12x}2\right)[/tex]

    Simplify:

    [tex]\dfrac1{16}-\dfrac1{16}\cos6x-\dfrac1{16}\cos12x+\dfrac1{16}\cos6x\cos12x[/tex]

    Angle sum identity for cosine:

    [tex]\dfrac1{16}-\dfrac1{16}\cos6x-\dfrac1{16}\cos12x+\dfrac1{16}\left(\dfrac{\cos6x+\cos18x}2\right)[/tex]

    Simplify:

    [tex]\dfrac1{16}-\dfrac1{32}\cos6x-\dfrac1{16}\cos12x+\dfrac1{32}\cos18x[/tex]
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