Trigonometric function based sums

Question

Find the value of $$\frac{1}{\sin45^{\circ}\sin46^{\circ}} + \frac{1}{\sin47^{\circ}\sin48^{\circ}}+\dots+\frac{1}{\sin133^{\circ}\sin134^{\circ}}.$$ The options given are $\sec 1, \csc 1,\cot 1$, none....please show the steps. I cannot understand this, please help me out.

Answer 1

This is what is sometimes called a "trick question." You do not need any advanced trigonometric identities to solve it.

Your choices are

1. $\sec1^\circ\approx1$
2. $\csc1^\circ=\csc\left(\frac{\pi}{180}\right)=\frac{1}{\sin\left(\frac{\pi}{180}\right)}\approx\frac{180}{\pi}<60$
3. $\cot1^\circ=\cot\left(\frac{\pi}{180}\right)=\frac{1}{\tan\left(\frac{\pi}{180}\right)}\approx\frac{180}{\pi}<60$
4. None

You are asked to add up 88 terms, each of which is larger than $1$.

Which is the correct answer?