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to prove that (r=1 to r=7) ∑tan 2 r∏/16=(r=1 to r=7)∑cot 2 r∏/16=35

to prove that


(r=1 to r=7)∑tan2r∏/16=(r=1 to r=7)∑cot2r∏/16=35

Grade:12

1 Answers

Nishant Vora IIT Patna
askIITians Faculty 2467 Points
6 years ago
Hello Student, Please find the solution

(r=1 to r=7)∑tan2r∏/16
= tan2∏/16 + tan22∏/16+ tan23∏/16+ tan24∏/16+ tan25∏/16+ tan26∏/16+ tan27∏/16
rearranging terms
=tan2∏/16+ tan27∏/16+ tan22∏/16+ tan26∏/16+ tan23∏/16tan25∏/16+ tan24∏/16
[Now, tan24∏/16 = tan2∏/4 = 1]
Also,

tan2∏/16+ tan27∏/16
= tan2∏/16+ cot2∏/16
=now you can convert to sine, cosine and find the ans

thanks,
All the best


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