Question

cos²1+cos²2+cos²3+......................+cos90​

Answer 1

Answer:

44.5

Hope this helps.

Step-by-step explanation:

Let C = cos² 1° + cos² 2° + ... + cos² 89° + cos² 90°

and S = sin² 1° + sin² 2° + ... + sin² 89° + sin² 90°.

Since cos² x + sin² x = 1, we get C + S = 90.

Since cos² x - sin² x = cos 2x, we get

C - S = cos 2° + cos 4° + ... + cos 88° + cos 90° + cos 92° + ... + cos 176° + cos 178° + cos 180°

Since cos (180° - x) = -cos x, this becomes

C - S = cos 2° + cos 4° + ... + cos 88° + cos 90° - cos 88° - ... - cos 4° - cos 2° - cos 0°

= cos 90° - cos 0° = 0 - 1 = -1

So we have C + S = 90 and C - S = -1.

Adding these gives 2C = 89  =>  C = 89/2 = 44.5