Prove sum of cosec values =0

C

Colin67

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Dear All

Apologies for posting two questions on the same day, I was really helped with the last question.

In this question has have answered parts a and b, I am struggling with part c.
cosec.jpg

The answer to part b is cot theta/2 -cot 4 theta. I am assuming this previous part will help in answering part c and have tried a similar approach but to no avail.
Am I missing something straightforward?
 
Colin67 said:
Dear All

Apologies for posting two questions on the same day, I was really helped with the last question.

In this question has have answered parts a and b, I am struggling with part c.
View attachment 16975

The answer to part b is cot theta/2 -cot 4 theta. I am assuming this previous part will help in answering part c and have tried a similar approach but to no avail.
Am I missing something straightforward?
Click to expand...
Did you try replacing \(\displaystyle \theta\) in the equation for (b), by \(\displaystyle \frac{4\pi}{15}\) ?
 
Hi
Do you not just end up with the equation given in c?
I've tried putting it in terms of cot θ in a similar approach to part b but get cot θ/2 -cot8θ
 
Colin67 said:
Hi
Do you not just end up with the equation given in c?
I've tried putting it in terms of cot θ in a similar approach to part b but get cot θ/2 -cot8θ
Click to expand...
Do you not just end up with the equation given in c? Yes partly! That is Subhotosh's point! They can't be the same result as the equation in part b has two terms while the eq in part c has three terms.
Can you please post your result for part b?
 
Subhotosh Khan said:
Did you try replacing \(\displaystyle \theta\) in the equation for (b), by \(\displaystyle \frac{4\pi}{15}\) ?
Click to expand...
1583272575447.png
So you found that you need to prove:

cot(2pi/15) - cot (16pi/15) + cosec(32pi/15) = 0

If I were to do this problem, I'll observe that

cot(16pi/15) = cot(pi/15) and

cosec(32 pi/15) = cosec (2 pi/15)

I'll substitute \(\displaystyle \frac{\pi}{15} \ \ by \ \ \phi \)

Then convert everything to sine and cosine and go.....
 
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