If x = m A + n A and y = m A + n A, then what is x² - y² equal to ?

Correct Answer is : m2−n2m² - n²m2−n2

Correct Answer is : m2−n2m² - n²m2−n2

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UPSC CDS 2 2021 Math Paper

Question : 56 of 100

Marks: +1, -0

x = m \sec A + n \tan A

y = m \tan A + n \sec A

x^{2} - y^{2}

equal to ?

Solution:

Concept:

Use the algebraic identity for the difference of squares and the trigonometric identity

\sec^2 A - \tan^2 A = 1

Explanation:

Start with

x = m\sec A + n\tan A

and

y = m\tan A + n\sec A

Square both expressions:

x^2 = m^2\sec^2 A + n^2\tan^2 A + 2mn\sec A\tan A

y^2 = m^2\tan^2 A + n^2\sec^2 A + 2mn\tan A\sec A

Now subtract

y^2

from

x^2

x^2 - y^2 = (m^2\sec^2 A + n^2\tan^2 A + 2mn\sec A\tan A) - (m^2\tan^2 A + n^2\sec^2 A + 2mn\tan A\sec A)

The cross terms

2mn\sec A\tan A

cancel.

Group like terms:

x^2 - y^2 = m^2(\sec^2 A - \tan^2 A) + n^2(\tan^2 A - \sec^2 A)

Factor

-1

from the second group:

n^2(\tan^2 A - \sec^2 A) = -n^2(\sec^2 A - \tan^2 A)

Thus

x^2 - y^2 = (m^2 - n^2)(\sec^2 A - \tan^2 A)

Using

\sec^2 A - \tan^2 A = 1

, we get

x^2 - y^2 = m^2 - n^2

Answer:

\boxed{m^2 - n^2}

(Option A)

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