Question

Find $3({\sin }^{4}x+{\cos }^{4}x)-2({\sin }^{6}x+{\cos }^{6}x)$

Answer 1

Let $y=-2({\sin }^{6}x+{\cos }^{6}x)+3({\sin }^{4}x+{\cos }^{4}x)$

We know that $({\sin }^{2}x+{\cos }^{2}x)=1$

$a^2+b^2=(a+b{)}^2-2aband{a}^3+b^3=(a+b{)}^3-3ab(a+b)$

Using these results we can reduce

$y=-2\left[\right(si{n}^{2}x+co{s}^{2}x{)}^3-3si{n}^{2}xco{s}^{2}x(si{n}^{2}x+co{s}^{2}x)]+3[(si{n}^{2}x+co{s}^{2}x{)}^2-2si{n}^{2}xco{s}^{2}x]$

$y=-2[1-3si{n}^{2}xco{s}^{2}x]+3[1-2si{n}^{2}xco{s}^{2}x]$

$y=1$