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Question
If \[\text{ sin } x + \text{ cos } x = a\], find the value of \[\left|\text { sin } x - \text{ cos } x \right|\] .
Solution
Given: \[\text{ sin } x + \text{ cos } x = a\]
Now,
\[\left( \text{ sin } x + \text{ cos } x \right)^2 + \left( \text{ sin } x - \text{ cos } x \right)^2 = \sin^2 x + \cos^2 x + 2\text{ sin } x\text{ cos } x + \sin^2 x + \cos^2 x - 2\text{ sin } x\text{ cos } x\]
\[ \Rightarrow \left( \text{ sin } x + \text{ cos } x \right)^2 + \left( \text{ sin } x - \text{ cos } x \right)^2 = 2\left( \sin^2 x + \cos^2 x \right)\]
\[ \Rightarrow \left( \text{ sin } x + \text{ cos } x \right)^2 + \left( \text{ sin } x - \text{ cos } x \right)^2 = 2\]
\[\therefore a^2 + \left( \text{ sin } x - \text{ cos } x \right)^2 = 2\]
\[ \Rightarrow \left( \text{ sin } x - \text{ cos } x \right)^2 = 2 - a^2 \]
\[ \Rightarrow \sqrt{\left( \text{ sin } x - \text{ cos } x \right)^2} = \sqrt{2 - a^2}\]
\[ \Rightarrow \left| \text{ sin } x - \text{ cos } x \right| = \sqrt{2 - a^2} \left( \sqrt{x^2} = \left| x \right| \right)\]
Thus, the required value is \[\sqrt{2 - a^2}\] .
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