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$I f \sin θ + \cos θ = a t h e n \sin^{4} θ + \cos^{4} θ =$

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$1 - \frac{1}{2} {(a^{2} - 1)}^{2}$

$1 - \frac{1}{2} {(a^{2} + 1)}^{2}$

$1 + \frac{1}{2} {(a^{2} + 1)}^{2}$

$1 + \frac{1}{2} {(a^{2} - 1)}^{2}$

answer is A.

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Detailed Solution

$sinθ + cosθ = a 1 + 2 sinθ cosθ = a^{2} 2 sinθ cosθ = a^{2} - 1 \sin^{4} θ + \cos^{4} θ = {(\sin^{2} θ + \cos^{2} θ)}^{2} - 2 \sin^{2} θ \cos^{2} θ = 1 - \frac{1}{2} {(a^{2} - 1)}^{2}$

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