. The value of cos2 5° plus cos2 10° plus cos2 15°


+......cos2 85° + cos2 90° is

A. 9 1/2

B. 9

C. 8 1/2

D. 8

 

Best Answer

Correct Option is : C

Solution :

Given cos2 5° + cos2 10° + cos2 15° + …. + cos2 85° + cos2 90°

We can rearrange the terms as (cos2 5° + cos2 85°) + ( cos2 10° + cos2 80°) - - - - - - - - - - - - - - - - - - - - - + cos2 45° + cos2 90°

As we know that sin2θ + con2θ = 1

But sinθ = cos (90 - θ)

⇒ sin2θ = con2 (90 - θ)

the ending terms in the series can be changed to

cos2 85° = cos2(90 - 5°) = sin25°

cos2 80° = cos2(90 - 10°) = sin210° and so on

therefore the given question changes to

(cos2 5° + cos2 85°) + ( cos2 10° + cos2 80°) - - - - - - - - - - - - - - - - - - - - - + cos2 45° + cos2 90° = (cos2 5° + sin2 5°) + ( cos2 10° + sin2 10°) - - - - - - - - - - - - - - - - - - - - - + cos2 45° + cos2 90°

= (1 + 1 + 1 - - - - - - - - ) 8 times + cos2 45° + cos2 90°

= 8 + (1/√2)2

= 8 + 1/2

= 17/2

= 8 1/2

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