Question

Evaluate ;
cos^2 O + cos^2 1 + cos^2 2 + .....cos^2 89
${ \cos(0) }^{2} + { \cos(1) }^{2} + { \cos(2) }^{2} + ................. { \cos(89) }^{2}$
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Answer 1

Answer:

we know that ,

sin²@ + cos²@ = 1

cos(90-@) = sin@

cos0° = 1

cos45° = 1/√2

use all in question now,

cos^2 O + cos^2 1 + cos^2 2 + .....cos^2 89

after 45° all we will change all into sin@

like cos²89° = cos²(90-1) = sin²1°

cos² 46° = cos²(90-44°) = sin²44°

so, all will change, than we will have series :-

cos^2 O + cos^2 1 + cos^2 2 + .....cos²45°.... sin²2 + sin²1

cos²0° + cos²45° + (cos²1°+sin²1°) + (cos²2+sin²2) + ____________ (sin²44+cos²44)

1+1/2 + 1 +1 ________________1 (total 44 terms makes 1)

= 1+1/2 + 44

= 45+1/2

= 45(1/2) (Ans.)