<s>[INST] Solve the equation $27 \sin(x) \cdot \cos^2(x) \cdot \tan^3(x) \cdot \cot^4(x) \cdot \sec^5(x) \cdot \csc^6(x) = 256$. 
Solve the equation $27 \sin(x) \cdot \cos^2(x) \cdot \tan^3(x) \cdot \cot^4(x) \cdot \sec^5(x) \cdot \csc^6(x) = 256$.

I was hoping some things would cancel out when I expanded this but nothing. I think using inequalities will help.
 [/INST] So after reducing the equation the equation will look like that:
$$27\csc^6(x)\sec^2(x)=256$$
OR:
$$\sin^6(x)\cos^2(x)=\frac{27}{256}$$
Then:
$$\sin^6(x)-\sin^8(x)=\frac{27}{256}$$
Or:
$$\sin^8(x)-\sin^6(x)+\frac{27}{256}=0$$
I found that $\sin^2(x)-\frac{3}{4}=0$ is divisible above equation.I there used the features of high degree polynomials.
So the equation will look like in this way:
$$(\sin^2(x)-\frac{3}{4})(\sin^6(x)+\frac{1}{4}\sin^4(x)+\frac{3}{4}\sin^2(x)+\frac{27}{64})=0$$
And:
$$\sin^2(x)-\frac{3}{4}=0$$ 
The rest is for you.
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