If $sin x + cos x = a,$ then $| sin x - cos x |$ equals

\sqrt{2 - a^{2}}

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\sqrt{2 + a^{2}}

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\sqrt{a^{2} - 2}

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\sqrt{a^{2} - 4}

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Solution

The correct option is C $\sqrt{2 - a^{2}}$
Given, $sin x + cos x = a$
${sin}^{2} x + {cos}^{2} x = 1$
${(sin x + cos x)}^{2} - 2 sin x cos x = 1$
$sin x + cos x = a$
Squaring on both sides,
${sin}^{2} x + {cos}^{2} x + 2 sin x c o s x = a^{2}$
$1 + 2 sin x cos x = a^{2}$
$2 sin x cos x = a^{2} - 1$
$= {(sin x + cos x)}^{2} - 2 (sin x cos x) = 1$
$= {(sin x + cos x)}^{2} + a^{2} - 1 = 1$
$= {(sin x + cos x)}^{2} = 2 - a^{2}$
$= | sin x - cos x | = \sqrt{2 - a^{2}}$

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