# Trigonometry - Cosec summation

Saitama

## Homework Statement

If
$$\csc\frac{\pi}{32}+\csc\frac{\pi}{16}+\csc\frac{\pi}{8}+\csc\frac{\pi}{4}+\csc\frac{\pi}{2}$$
has the value equal to ##\cot\frac{\pi}{A}## then find A.
A)61
B)62
C)63
D)64

## The Attempt at a Solution

Writing cosec in terms of sin and taking the LCM to make a common denominator doesn't seem to be of any help.

I can find the value of each term but that would be tedious and of no use.

I honestly cannot figure out how should I proceed here.

Any help is appreciated. Thanks!

Last edited:

Homework Helper

## Homework Statement

If
$$\csc\frac{\pi}{32}+\csc\frac{\pi}{16}+\csc\frac{\pi}{2}+\csc\frac{\pi}{4}+\csc\frac{\pi}{2}$$
has the value equal to ##\cot\frac{\pi}{A}## then find A.
A)61
B)62
C)63
D)64

## The Attempt at a Solution

Writing cosec in terms of sin and taking the LCM to make a common denominator doesn't seem to be of any help.

I can find the value of each term but that would be tedious and of no use.

I honestly cannot figure out how should I proceed here.

Any help is appreciated. Thanks!

Try showing cot(x)-cot(2x)=csc(2x).

• 1 person
Homework Helper
##\csc(\frac{\pi}{2})## appears twice.
All the angles are successive doublings of ##\pi/32##

Saitama
Try showing cot(x)-cot(2x)=csc(2x).

Wow! Thanks a lot Dick! How did you come up with that?

##\csc(\frac{\pi}{2})## appears twice.
All the angles are successive doublings of ##\pi/32##
Very sorry for the typo, its ##\pi/8## instead of the second ##\pi/2##.

Homework Helper
Wow! Thanks a lot Dick! How did you come up with that?

I guessed the series must telescope somehow. So somehow cot(2x) must be related to cot(x) with the difference related to a csc. Seems obvious in retrospect, yes?

Last edited:
Saitama
Seems obvious in retrospect, yes?

Yes. I liked the way you came up with cot(x)-cot(2x) and solved the problem in few seconds where I was stuck for a week.

Thank you again! :)

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