Value of cosec frac{π }{{18}} - √ 3 ,sec, frac{π }{{18}} is a

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3.Trigonometrical Ratios, Functions and Identities

normal

The value of $cosec \frac{\pi }{{18}} - \sqrt 3 \,sec\, \frac{\pi }{{18}}$ is a

surd

rational which is not integral

negative natural number

natural number

Solution

$\frac{1}{{\sin \pi /18}}\,\, – \,\,\frac{{\sqrt 3 }}{{\cos \pi /18}}$ $=$$\frac{{2\,\left[ {\frac{1}{2}\cos \frac{\pi }{{18}}\, – \,\frac{{\sqrt 3 }}{2}\,\,\sin \frac{\pi }{{18}}} \right]}}{{\frac{{\sin \frac{\pi }{9}}}{2}}}$

$=$$\frac{{4\left[ {\sin \frac{\pi }{6}\,\,\cos \frac{\pi }{{18}}\,\, – \,\,\cos \frac{\pi }{6}\,\,\sin \frac{\pi }{{18}}} \right]}}{{\sin \frac{\pi }{9}}}$ $= 4$

Standard 11

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