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SSC CPO SI and ASI Model Paper 7

Question : 125 of 200

Marks: +1, -0

\cos^2 1^{\circ} + \cos^2 3^{\circ} + \cos^3 5^{\circ} + \cos^2 7^{\circ} + \dots + \cos^2 85^{\circ} + \cos^2 87^{\circ} + \cos^2 89^{\circ}

is equal to

Solution:

(1)

\cos^2 1^{\circ} + \cos^2 3^{\circ} + \cos^3 5^{\circ} + \cos^2 7^{\circ} + \dots + \cos^2 85^{\circ} + \cos^2 87^{\circ} + \cos^2 89^{\circ}

In the given expression, we can seethat the angles are 1°, 3°, 5°, 7°,9°, …, 81°, 83°, 85°, 87°, 89°

Let the number of terms in theexpression be n, d be the commondifference, and tn be the nth angle.

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

\Rightarrow 1 + (n - 1)2 = 89

⇒n = 45

\therefore \cos^2 1^{\circ} + \cos^2 3^{\circ} + \cos^3 5^{\circ} + \cos^2 7^{\circ} + \dots + \cos^2 85^{\circ} + \cos^2 87^{\circ} + \cos^2 89^{\circ}

= \cos^2 1^{\circ} + \cos^2 89^{\circ} + \cos^2 3^{\circ} + \cos^2 87^{\circ} + \cos^2 5^{\circ} + \cos^2 85^{\circ} + \dots + \cos^2 41^{\circ} + \cos^2 49^{\circ} + \cos^2 43^{\circ} + \cos^2 47^{\circ} + \cos^2 45^{\circ}

= \cos^2 1^{\circ} + \sin^2 1^{\circ} + \cos^2 3^{\circ} + \sin^2 3^{\circ} + \dots + \cos^2 43^{\circ} + \sin^2 43^{\circ} + \cos^2 45^{\circ}

= \left[ 1 + 1 + 1 + \dots + 1 \right]_{22 \text{ times}}

+ \left[ \frac{1}{\sqrt{2}} \right]^2

= 45/2

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