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UPSC CDS I 2019 Math Paper

Question : 58 of 100

Marks: +1, -0

\sin^2 25^{\circ} + \sin^2 65^{\circ}

Solution:

Concept:

Use the complementary angle identity and the Pythagorean identity.

Explanation:

We want to find

\sin^2 25^{\circ} + \sin^2 65^{\circ}

Notice that

25^{\circ} + 65^{\circ} = 90^{\circ}

, so they are complementary angles.

For complementary angles,

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

Thus

\sin 65^{\circ} = \sin(90^{\circ} - 25^{\circ}) = \cos 25^{\circ}

Squaring both sides gives

\sin^2 65^{\circ} = \cos^2 25^{\circ}

Now substitute:

\sin^2 25^{\circ} + \sin^2 65^{\circ} = \sin^2 25^{\circ} + \cos^2 25^{\circ}

The Pythagorean identity states

\sin^2 \theta + \cos^2 \theta = 1

for any angle

\theta

Applying it with

\theta = 25^{\circ}

, we get

\sin^2 25^{\circ} + \cos^2 25^{\circ} = 1

Therefore, the sum equals

1

Answer:

1

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