Find the value of
$$\frac{\sin^{6}15^{\circ} + \sin^{6}75^{\circ} + 6\sin^{2}15^{\circ}\sin^{2}75^{\circ}}{\sin^{4}15^{\circ} + \sin^{4}75^{\circ} + 5\sin^{2}15^{\circ}\sin^{2}75^{\circ}}$$
$$Let\ \sin^215^{\circ}\ =a\ and\ \sin^275^{\circ}\ =\ b\ $$
Then the given equation becomes,
$$\ \frac{\ a^3+b^3+6ab}{a^2+b^2+5ab}$$
=$$\ \frac{\ \left(a+b\right)^3-3ab\left(a+b\right)+6ab}{\left(a+b\right)^2-2ab+5ab}$$ $$\ \therefore\ \ a^3+b^3=\left(a+b\right)^3-3ab\left(a+b\right)$$ , $$\ \therefore\ \ a^2+b^2=\left(a+b\right)^2-2ab$$
Using $$\sin\left(\theta\right)\ =\cos\left(90-\theta\ \ \right)$$
we get, $$\sin^275^{\circ\ }=\cos^215^{\circ\ }$$
now, a+b = $$\sin^215^{\circ\ }+\sin^275^{\circ\ }$$ = $$\sin^215^{\circ\ }+\cos^215^{\circ\ }$$ = 1.
The equation becomes, $$=\ \frac{\ 1-3ab+6ab}{1-2ab+5ab}\ =\ \frac{\ 1+3ab}{1+3ab}\ =1\ $$
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