If x a cos b sin and y a sin minus b cos then what is x2 y2 equal to ........

Quantitative Aptitude (CDS) Trignometry

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If x = a cos $\Theta$ + b sin $\Theta$ and y = a sin $\Theta$ − b cos $\Theta$ , then what is x² + y² equal to?

2ab

a+b

a²+b²

a²-b²

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Explanation:

As,
x = a cos $\Theta$ + b sin $\Theta$ ……….. (i)
y = a sin $\Theta$ − b cos $\Theta$ ………… (ii)
Squaring and adding (i) and (ii)
x² + y² = (a cos $\Theta$ + b sin $\Theta$ )² +(a cos $\Theta$ − b sin $\Theta$ )²
= a² cos² $\Theta$ + b² sin² $\Theta$ + 2ab sin $\Theta$ . cos $\Theta$ + a² sin² $\Theta$ + b² cos² $\Theta$ − 2ab sin $\Theta$ . cos $\Theta$
= a²(cos² $\Theta$ + sin² $\Theta$ ) + b²(sin² $\Theta$ + cos² $\Theta$ )
= a² × 1 + b² × 1
= a² + b²