Question

Sin^2 1°+sin^2 3°+sin2^5°+........+sin^2 85°+sin^2 87°+sin^2 89° =?​

Answer 1

=Sin²1° + sin² 3° + sin² 5° +........ + sin² 85° + sin² 87° + sin² 89°

= [sin (90°-89°)]² + [sin (90°-87°)]² + [sin (90°-85°)]² +.....+sin²45°+...... + sin² 85° + sin² 87° + sin² 89°

= cos²89°+ cos²87°+ cos²85° +....... + sin²45°+....... +sin² 85° + sin² 87° + sin² 89°

Since sin²A + cos² A = 1

= cos²89° + sin² 89° + cos²87°+ sin² 87°+ cos²85°+ sin² 85°+....... +sin²45°

= 1 + 1 +1 +........ +1/2

Since there are 45 terms in the series,

There are 13 pairs and 1 unpaired.

Hence the sum is:

= 13 + 1/2

=26/2 + 1/2

= 27/2

Answer 2

Answer: 45/2

Step-by-step explanation:

Sin^2 1° + sin^2 89° = {cos (90-89°} ^2 + sin^2 89° = cos^2 89° + sin^2 89° = 1

Simultaneously there are 45 terms in total from 1 to 89, so 22 pairs of cos and sin making value of 22, and 1 unpaired that is sin^2 45° = 1/2.

So the ans is 22+ 1/2 = 45/2..

Thankyou