A solution ( x , y ) of the system of equations x − y = 3 1 and cos 2 ( π x ) − sin 2 ( π y ) = 2 1 is

cos 2 ( π x ) − sin 2 ( π y ) = cos { π ( x + y )} cos { π ( x − y )} = 2 1 and cos 3 π = 2 1 , so cos { π ( x + y )} = 1 ⇒ x + y = 2 n , n ∈ Z . Options A, C and D satisfy both conditions; in B, x + y = 1 is odd.

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MathematicsTrigonometric Ratios & IdentitiesMiscellaneous/MixedMedium2 minQB

MathematicsMediummultiple choice

A solution $(x, y)$ of the system of equations $x - y = \frac{1}{3}$ and $cos^{2} (π x) - sin^{2} (π y) = \frac{1}{2}$ is

Options:(select one or more)

Answer:

A, C, D

Solution:

$cos^{2} (π x) - sin^{2} (π y) = cos {π (x + y)} cos {π (x - y)} = \frac{1}{2}$ and $cos \frac{π}{3} = \frac{1}{2}$ , so $cos {π (x + y)} = 1 \Rightarrow x + y = 2 n$ , $n \in Z$ . Options A, C and D satisfy both conditions; in B, $x + y = 1$ is odd.

Stream:JEESubject:MathematicsTopic:Trigonometric Ratios & IdentitiesSubtopic:Miscellaneous/Mixed

⏱ 2mℹ️ Source: QB

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