Question

Solve:
${\left(\frac{{\sin }^{m}x}{{\sin }^{n}x}\right)}^{m+n}.{\left(\frac{{\sin }^{n}x}{{\sin }^{q}x}\right)}^{n+q}{\left(\frac{{\sin }^{q}x}{{\sin }^{m}x}\right)}^{q+m}$

Answer 1

Let

$P={\left(\frac{{\sin }^{m}x}{{\sin }^{n}x}\right)}^{m+n}{\left(\frac{{\sin }^{n}x}{{\sin }^{q}x}\right)}^{n+q}{\left(\frac{{\sin }^{q}x}{{\sin }^{m}x}\right)}^{q+m}$

$P={\left(\sin x\right)}^{(m-n)(m+n)}.{\left(\sin x\right)}^{(n-q)(n+q)}.{\left(\sin x\right)}^{(q-m)(q+m)}$

$P={\left(\sin x\right)}^{m^2-n^2+n^2-q^2+q^2-m^2}={\left(\sin x\right)}^0=1$