Question

cosec10-√3sec10
value

Answer 1

 ( csc 10 - √3. sec 10 ) = (1/ sin 10) - (√3/ cos 10) = ( cos 10 - √3. sin 10 ) / ( sin 10 cos 10 ) = 4 [ (1/2) cos 10 - (√3 /2) sin 10 ] / ( 2 sin 10 cos 10 ) = 4 [ sin 30 cos 30 - cos 30 sin 10 ] / [ sin 2(10) ] = 4 [ sin (30-10) ] / ( sin 20 ) = 4 .

Answer 2

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$\csc(10) - \sqrt{3} \sec(10) \\ = \frac{1}{ \sin(10) } - \frac{ \sqrt{3} }{ \cos(10) } \\ = \frac{ \cos(10) - \sqrt{3} \sin(10) }{ \sin(10) \cos(10) } \\ = 4( \frac{ \frac{1}{2} \cos(10) - \frac{ \sqrt{3} }{2} \sin(10) }{2 \sin(10) \cos(10) } ) \\ = 4( \frac{ \sin(30) \cos(10) - \cos(30) \sin(10) }{ \sin2(10) } ) \\ = 4( \frac{ \sin(30 - 10) }{ \sin(20) } ) = 4 \frac{ \sin(20) }{ \sin(20) } \\ = 4$
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