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SSC CPO SI and ASI Model Paper 4

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Question : 134 of 200

Marks: +1, -0

Find the value of

\cos^2 1^{\circ} + \cos^2 5^{\circ} + \cos^2 9^{\circ} + \dots + \cos^2 85^{\circ} + \cos^2 89^{\circ}.

$\frac{45}{2}$
$\frac{23}{2}$
$\frac{37}{2}$
$\frac{59}{2}$

Solution:

(2)

\cos^2 1^{\circ} + \cos^2 5^{\circ} + \cos^2 9^{\circ} + \dots + \cos^2 85^{\circ} + \cos^2 89^{\circ}

In the given expression we can seethat the angles are 1° , 5° , 9° , 13° ,…, 81° , 85° , 89°

Let the number of terms in theexpression be n.

\therefore a + (n - 1)d = t_n

⇒ 1 + (n – 1)4 = 89

⇒ n = 23

\therefore \cos^2 1^{\circ} + \cos^2 5^{\circ} + \cos^2 9^{\circ} + \dots + \cos^2 85^{\circ} + \cos^2 89^{\circ}

°

= \cos^2 1^{\circ} + \cos^2 89^{\circ} + \cos^2 5^{\circ} + \cos^2 85^{\circ} + \dots + \cos^2 37^{\circ} + \cos^2 53^{\circ} + \cos^2 41^{\circ} + \cos^2 49^{\circ} + \cos^2 45^{\circ}

= \cos^2 1^{\circ} + \sin^2 1^{\circ} + \cos^2 5^{\circ} + \sin^2 5^{\circ} + \dots + \cos^2 41^{\circ} + \sin^2 41^{\circ} + \cos^2 45^{\circ}

= [1 + 1 + 1 + \dots + 1]_{\text{(11 times)}} + \left(\frac{1}{\sqrt{2}}\right)^2

= \frac{23}{2}

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