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Answer 1

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Answer 2

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ACCORDING TO QUESTION

$\frac{ \sin(30) + \tan(45) - \csc(60) }{ \sec(30) + \cos(60) + \cot(45) } \\ = \frac{ \frac{1}{2} + 1 - \frac{2}{ \sqrt{3} } }{ \frac{2}{ \sqrt{3} } + \frac{1}{2} + 1} \\ = \frac{ \frac{ \sqrt{3} + 2 \sqrt{3} - 4}{2 \sqrt{3} } }{ \frac{4 + \sqrt{3} + 2 \sqrt{3} }{2 \sqrt{3} } } \\ = \frac{3 \sqrt{3} - 4 }{3 \sqrt{3} + 4} \\ by \: rationalization \\ = \frac{(3 \sqrt{3} - 4)(3 \sqrt{3} - 4) }{(3 \sqrt{3} + 4)(3 \sqrt{3} - 4) } \\ = \frac{ {(3 \sqrt{3}) }^{2} + {(4) }^{2} - 2(3 \sqrt{3} )(4) }{ {(3 \sqrt{3} }^{2}) - {(4)}^{2} } \\ = \frac{27 + 16 - 24 \sqrt{3} }{27 - 16} \\ = \frac{43 - 24 \sqrt{3} }{11}$