If $$15 \sin^4 \alpha + 10 \cos^4 \alpha = 6$$, for some $$\alpha \in R$$, then the value of $$27 \sec^6 \alpha + 8\operatorname{cosec}^6 \alpha$$ is equal to :

We are given $$15\sin^4\alpha + 10\cos^4\alpha = 6$$. Let $$\sin^2\alpha = s$$ and $$\cos^2\alpha = 1 - s$$, so the equation becomes $$15s^2 + 10(1-s)^2 = 6$$, which expands to $$15s^2 + 10 - 20s + 10s^2 = 6$$, giving $$25s^2 - 20s + 4 = 0$$. This factors as $$(5s - 2)^2 = 0$$, so $$s = \frac{2}{5}$$, meaning $$\sin^2\alpha = \frac{2}{5}$$ and $$\cos^2\alpha = \frac{3}{5}$$.

Now we compute $$27\sec^6\alpha + 8\csc^6\alpha = \frac{27}{\cos^6\alpha} + \frac{8}{\sin^6\alpha} = \frac{27}{(3/5)^3} + \frac{8}{(2/5)^3} = \frac{27 \cdot 125}{27} + \frac{8 \cdot 125}{8} = 125 + 125 = 250$$.