The value of ^2 5^ + ^2 10^ + ^2 15+ . .+ ^2 85^ + ^2 90^ =

Step-by-step solution

Evaluating the series ^2 5^ + ^2 10^ + ^2 15^ + + ^2 85^ + ^2 90^ . The angles form an arithmetic sequence with 18 terms from 5^ to 90^ in increments of 5^ . Using the identity ^2 + ^2 (90^ - ) = ^2 + ^2 = 1 , pair terms symmetrically about the center: ^2 5^ + ^2 85^ = 1 ^2 10^ + ^2 80^ = 1 ^2 15^ + ^2 75^ = 1 ^2 20^ + ^2 70^ = 1 ^2 25^ + ^2 65^ = 1 ^2 30^ + ^2 60^ = 1 ^2 35^ + ^2 55^ = 1 ^2 40^ + ^2 50^ = 1 These eight pairs contribute 8 1 = 8 to the sum. The remaining terms ^2 45^ = ( 1 2 )^2 = 1 2 and ^2 90^ = 1