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The Value of 2 ( Sin 2 X + 2 Cos 2 X − 1 ) Cos X − Sin X − Cos 3 X + Sin 3 X is - Mathematics

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Question

The value of \[\frac{2\left( \sin 2x + 2 \cos^2 x - 1 \right)}{\cos x - \sin x - \cos 3x + \sin 3x}\] is 

 

Options

  •  cos x

  • sec x

  •  cosec x

  • sin x

MCQ
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Solution

cosec 

\[\text{ We have } , \]

\[\frac{2\left( \sin2x + 2 \cos^2 x - 1 \right)}{\text{ cos } x - \text{ sin } x - \cos3x + \sin3x}\]

\[ = \frac{2\left( \sin2x + \cos2x \right)}{\text{ cos } x - \text{ sin } x - 4 \cos^3 x + 3\text{ cos } x + 3\text{ sin } x - 4 \sin^3 x}\]

\[ = \frac{2\left( \sin2x + \cos2x \right)}{4\text{ cos } x - 4 \cos^3 x + 2\text{ sin } x - 4 \sin^3 x}\]

\[ = \frac{2\left( \sin2x + \cos2x \right)}{4\text{ cos } x\left( 1 - \cos^2 x \right) + 2\text{ sin } x\left( 1 - 2 \sin^2 x \right)}\]

\[ = \frac{2\left( \sin2x + \cos2x \right)}{4\text{ cos } x \sin^2 x + 2\text{ sin } x \cos2x}\]

\[ = \frac{2\left( \sin2x + \cos2x \right)}{2 \times 2\text{ sin } x \text{ cos } x \text{ sin } x + 2\text{ sin } x \cos2x}\]

\[ = \frac{2\left( \sin2x + \cos2x \right)}{2\sin2x \text{ sin } x + 2\text{ sin } x \cos2x}\]

\[ = \frac{2\left( \sin2x + \cos2x \right)}{2\text{ sinx } \left( \sin2x + \cos2x \right)}\]

\[ = \frac{1}{\text{ sin } x}\]

\[ = \text{ cosec } x \]

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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 9.5 [Page 44]

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RD Sharma Mathematics [English] Class 11
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.5 | Q 23 | Page 44

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