Question

sec2 − sin2 −2 sin4 2 cos4 −cos2 = 1

Answer 1

We have to find the sum of the series (sin 1)^2 + (sin 2)^2 +...+ (sin 90)^2
(sin 1)^2 + (sin 2)^2 +...+ (sin 90)^2
=> (sin 1)^2 + (sin 2)^2 +...+ (sin 90)^2
=> (sin 1)^2 + (sin 2)^2 +... + (sin 44)^2 + (sin 45)^2 + (sin 46)^2 ...+ (sin 90)^2
=> (sin 1)^2 + (sin 2)^2 +... + (sin 44)^2 + (sin 45)^2 + (sin 46)^2 ...+ (sin 90)^2
use sin (90 - x) = cos x
=> (sin 1)^2 + (sin 2)^2 +... + (sin 44)^2 + (sin 45)^2 + (sin(90 - 44))^2 ...+ (sin (90 - 0))^2
=> (sin 1)^2 + (sin 2)^2 +... + (sin 44)^2 + (sin 45)^2 + (cos 44)^2 + (cos 43)^2 ...+ (cos 0)^2
=> (sin 1)^2 + (cos 1)^2 + (sin 2)^2 + (cos 2)^2 +...+ (sin 44)^2 + (cos 44)^2 + (sin 45)^2 + (cos 0)^0
=> 44*1 + (1/sqrt 2)^2 + 1
=> 44 + 1/2 + 1
=> 45.5
The required sum is 45.5