First slide

Trigonometric Identities

Question

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

Moderate

A

$\sqrt{1 - a^{2}}$
B

$\sqrt{a^{2} - 1}$
C

$\sqrt{2 - a^{2}}$
D

$\sqrt{a^{2} - 2}$

Solution

$\sin x + \cos x = a \Rightarrow 1 + 2 \sin x \cos x = a^{2} \Rightarrow 2 \sin x \cos x = a^{2} - 1 ∴ |\sin x - \cos x| = \sqrt{{(\sin x - \cos x)}^{2}} = \sqrt{1 - 2 \sin x \cos x} = \sqrt{1 - (a^{2} - 1)} = \sqrt{2 - a^{2}}$

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