Prove that: <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c <mi mathvarian

lobht98

lobht98

Answered question

2022-06-26

Prove that:
c o s e c α 8 + c o s e c α 4 + c o s e c α 2 = cot α 16 cot α 2

Answer & Explanation

lilao8x

lilao8x

Beginner2022-06-27Added 22 answers

lternatively, one may use
csc α 2 = cot α cot α 2
giving
csc α 2 n = cot α 2 n + 1 cot α 2 n , n = 1 , 2 , 3 , ,
then summing from n=1 to n=3 and using a telescoping sum gives
csc α 8 + csc α 4 + csc α 2 = cot α 16 cot α 2 .
Sarai Davenport

Sarai Davenport

Beginner2022-06-28Added 4 answers

Note that in general
csc x + cot x = 1 + cos x sin x = cot 2 x .
Now, moving cot α 2 to the LHS and applying this identity multiple times gives
csc α 8 + csc α 4 + csc α 2 + cot α 2 = csc α 8 + csc α 4 + cot α 4 = csc α 8 + cot α 8 = cot α 16 .

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?