In which one of the following intervals the inequality  sinx < cosx < tanx < cotx can hold good?

In the second quadrant, sinx ( cosx is false, as sinx is positive and cosx is negative. 

In the fourth quadrant, cosx <tanx is false, as cosx is positive and tanx is negative.

In the third quadrant, i.e. $\left(\frac{5\mathrm{\pi }}{4},\frac{3\mathrm{\pi }}{2}\right)\text{ if }\tan \mathrm{x}<\cot \mathrm{x}$  then  ${\tan }^{2}x<1$ which is false.

Now, sinx < cosx is true in $\left(0,\frac{\pi }{4}\right)$ and tanx < cotx is also true,
 Further cosx < cotx, as cotx $=\frac{\cos \mathrm{x}}{\sin \mathrm{x}}$ and sinx < 1.