Question

Evaluate sin square 1 Degree + sin square 23 degree + sin square 45 degree + so on till + sin square 89 degree​

Answer 1

Given→

sin²1° + sin²23° + sin² 45 ° + sin² 67° + sin² 89°

Because there is a difference of 22° among all terms .

Solution→

Sin²1 ° can be written in the form of cos theta as

sin theta = cos (90- theta )

sin ²1. = cos² ( 90-1 )

Sin²1°. = cos²89°

Similarly →

sin²23°. = cos²( 90- 23)

sin²23°. = cos²67°

Now putting theses values in given series.

(cos²89° + sin²89°) +(cos²67° +sin²67°) +sin²45°

we know that sin²a + cos²a = 1 and value of sin 45° = 1/√2

→ 1 + 1 +( 1/√2 )²

→ 2 + 1/2

→ 5 /2