[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]
