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Thursday, March 26, 2015

2015/023) show that \tan\, 9^{\circ} - \tan\, 27^{\circ}- \tan\, 63^{\circ} + \tan\, 81^{\circ}= 4


we have \tan\, x + \cot\, x = \tan\, x + \dfrac{1}{\tan\, x}
 = \dfrac{tan ^2 x + 1}{\ tan\, x} = \dfrac{sec^2 x}{\tan,x}= \dfrac{1}{\cos\,x\sin\,x}=\dfrac{2}{2\cos\,x\sin\,x} = \dfrac{2}{\sin 2x} = 2 \csc 2x

 so
 \tan\, 9^{\circ} + \tan\, 81^{\circ} = 2 \csc\, 18^{\circ}
\tan\, 27^{\circ} + \tan\, 63^{\circ} = 2 \csc\,  54^{\circ}

so given value

= \tan\, 9^{\circ} - \tan\, 27^{\circ}- \tan\, 63^{\circ} + \tan\, 81^{\circ}=2( \csc\, 18^{\circ} - \csc\, 54^{\circ}) 

= 2(\dfrac{1}{\sin\,18^{\circ}} - \dfrac{1}{\sin\, 54^{\circ}})

= 2 \dfrac{2 \cos\, 36^{\circ}  \sin\, 18^{\circ}}{ \sin \,18^{\circ}  \sin \,54^{\circ} }
4

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