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UPSC CDS 2 2022 Math Paper

Question Numbers: 43-45

Consider the following for the next three (03) items that follow:

A triangle ABC with sides A

B = 15

cm, B

C = 9

cm, C

A = 12

cm is inscribed in a circle.

Question : 43 of 100

Marks: +1, -0

What is

\cos^{2} A + \cos^{2} B + \cos^{2} C

equal to ?

$\frac{3}{4}$
1
$\frac{5}{4}$
2

Validate

Solution:

Concept:

In a right-angled triangle, using complementary angle identities and the Pythagorean identity, the sum of squares of cosines of all angles simplifies to 1.

Explanation:

Given side lengths:

AB = 15

cm,

BC = 9

cm,

AC = 12

cm.

Check Pythagoras:

12^2 + 9^2 = 144 + 81 = 225 = 15^2

Thus,

\text{triangle } ABC

is right-angled at

C

, so

\angle C = 90^{\circ}

From angle sum:

\angle A + \angle B = 90^{\circ}

, so

\angle B = 90^{\circ} - \angle A

Then:

\cos^{2} A + \cos^{2} B + \cos^{2} C = \cos^{2} A + \cos^{2}(90^{\circ} - A) + \cos^{2} 90^{\circ}

Using

\cos(90^{\circ} - A) = \sin A

and

\cos 90^{\circ} = 0

\cos^{2} A + \sin^{2} A + 0

Apply identity

\sin^{2} A + \cos^{2} A = 1

: the sum equals

1

Answer:

1

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