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If sin x + cos x = a , find the value of | sin x − cos x |.

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Question

If  \[\text{ sin } x + \text{ cos } x = a\], find the value of \[\left|\text { sin } x - \text{ cos } x \right|\] .

 

 

Short/Brief Note
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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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Values of Trigonometric Functions at Multiples and Submultiples of an Angle
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Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle - Exercise 9.4 [Page 42]

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R.D. Sharma Mathematics [English] Class 11
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.4 | Q 13 | Page 42

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