to prove that

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

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