Question

The value of $co{s}^2{5}^{\circ }+co{s}^2{10}^{\circ }+co{s}^2{15}^{\circ }+\dots \dots \dots ..+co{s}^2{85}^{\circ }+co{s}^2{90}^{\circ }$ is equal to

• A. 10
• B. 19/2
• C. 9/2
• D. 17/2

Answer 1

The correct option is D 17/2

We know that, $sin({90}^{\circ }-\theta )=cos\theta$

Therefore $co{s}^2{5}^{\circ }+co{s}^2{10}^{\circ }+co{s}^2{15}^{\circ }+\dots \dots \dots ..+co{s}^2{85}^{\circ }+co{s}^2{90}^{\circ }$
= $si{n}^2(90-5{)}^{\circ }+si{n}^2(90-10{)}^{\circ }+si{n}^2(90-15{)}^{\circ }+\dots \dots \dots ..+co{s}^2{85}^{\circ }+co{s}^2{90}^{\circ }$

The above expression has (90/5) = 18 terms out of which $cos{90}^{\circ }$ is 0, leaving 17 terms.

Also, $si{n}^2\theta +co{s}^2\theta =1$

$si{n}^2(90-5{)}^{\circ }+si{n}^2(90-10{)}^{\circ }+si{n}^2(90-15{)}^{\circ }+\dots \dots \dots ..+co{s}^2{85}^{\circ }+0$
= $si{n}^2{85}^{\circ }+si{n}^2{80}^{\circ }+si{n}^2{75}^{\circ }+\dots \dots \dots ..+co{s}^2{75}^{\circ }+co{s}^2{85}^{\circ }$
Out of these 17 terms, there are 8 pairs which are of the form $si{n}^2\theta +co{s}^2\theta =1$; left out term is $co{s}^2{45}^{\circ }$ = $\frac{1}{2}$
= 8*1 + 0 + $\frac{1}{2}$
= 8 + $\frac{1}{2}$
= $\frac{17}{2}$

The value of $co{s}^2{5}^{\circ }+co{s}^2{10}^{\circ }+co{s}^2{15}^{\circ }+\dots \dots \dots ..+co{s}^2{85}^{\circ }+co{s}^2{90}^{\circ }$ is equal to 17/2