Definite Integration, keep getting wrong answer. 
Correct to 4 significant figures
  $$\int_{1}^{2}{\csc^24tdt}$$

Done this multiple times now and can't seem to get the answer at the back of the book.
Here's my attempt:
$$\int_{1}^{2}{\csc^24tdt}$$
$$-\dfrac{1}{4}\left[\cot 4(2)-\cot 4(1)\right]_{1}^{2}$$
$$-\dfrac{1}{4}\left[\frac{1}{\tan 8}-\frac{1}{\tan4}\right]$$
$$-\dfrac{1}{4}\left[-0.1471-0.8637\right]$$
$$-\dfrac{1}{4}\left[-1.0108\right]$$
$$0.2527$$
Back of the book has a similar answer which is $0.2572$
Where am I going wrong?

The book is wrong. Plain and simple. They transposed the last two digits, which is a simple typo. Originally, I was going to write the following, and I include it here for numerical reference.
You need to do all the rounding right at the end. Since we have calculators today, which usually carry 12-16 digits of precision, round-off error is the least of our worries. If you carry out the calculations without intermediate rounding, you get an answer of: $0.2527$.
As an aside, you cannot evaluate the integral directly, since it passes through a discontinuity at $\frac{\pi}{2}$, as shown below:

