Question

5sin30°+cos²45°-4tan²30°/2sin30°×cos30°+tan45°




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Answer 1

STEP-BY-STEP EXPLANATION:

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$\longrightarrow \frac{5 \sin {30}^{ \circ} + { \cos}^{2} {45}^{ \circ} - 4 { \tan}^{2} {30}^{ \circ} }{2 \sin {30}^{ \circ}. \cos {30}^{ \circ} + \tan {45}^{ \circ} }$

$\sf As \: we \: know \: that, \begin{cases} \sin {30}^{ \circ} = \frac{1}{ \sqrt{2} } \\ \\ \cos {30}^{ \circ} = \frac{ \sqrt{3} }{2} \\ \\ \cos {45}^{ \circ} = \frac{1}{ \sqrt{2} } \\ \\ \tan {30}^{ \circ} = \frac{1}{ \sqrt{3} } \\ \\ \tan {45}^{ \circ} = 1 \end{cases}$

$\longrightarrow \frac{5 \sin {30}^{ \circ} + { \cos}^{2} {45}^{ \circ} - 4 { \tan}^{2} {30}^{ \circ} }{2 \sin {30}^{ \circ}. \cos {30}^{ \circ} + \tan {45}^{ \circ} }$

$\longrightarrow \frac{5 ( \frac{1}{ \sqrt{2} } ) + { ( \frac{1}{ \sqrt{2} }) }^{2} - 4 {( \frac{1}{ \sqrt{3} } )}^{2} }{2 ( \frac{1}{ \sqrt{2} }) \times \frac{ \sqrt{3} }{2} + 1 }$

$\longrightarrow \frac{ \frac{5}{ \sqrt{2} } + \frac{1}{ {2} } - 4 (\frac{1}{ {3} } ) }{ \frac{2}{ \sqrt{2} } \times \frac{ \sqrt{3} }{2} + 1 }$

$\longrightarrow \frac{ \frac{5}{ \sqrt{2} } \times \frac{ \sqrt{2} }{ \sqrt{2} } + \frac{1}{ {2} } - 4 (\frac{1}{ {3} } ) }{ \frac{ \cancel{2}}{ \sqrt{2} } \times \frac{ \sqrt{3} }{ \cancel{2}} + 1 }$

$\longrightarrow \frac{ \frac{5}{ \sqrt{2} } \times \frac{ \sqrt{2} }{ \sqrt{2} } + \frac{1}{ {2} } - \frac{4}{ {3} } }{ \frac{ \cancel{2}}{ \sqrt{2} } \times \frac{ \sqrt{3} }{ \cancel{2}} + 1 }$

$\longrightarrow \frac{ \frac{5 \sqrt{2} }{ {2} } + \frac{1}{ {2} } - \frac{4}{ {3} } }{ \frac{ \sqrt{3} }{ \sqrt{2} } + 1 }$

$\longrightarrow \frac{ \frac{5 \sqrt{2} }{ {2} } + \frac{1}{ {2} } - \frac{4}{ {3} } }{ \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{2} } }$

$\longrightarrow [ \frac{5 \sqrt{2} + 3 - 8}{6} ] \times \frac{ \sqrt{2} }{ \sqrt{3} + \sqrt{2} }$

$\longrightarrow [ \frac{5 \sqrt{2} - 5}{6} ] \times \frac{ \sqrt{2} }{ \sqrt{3} + \sqrt{2} }$

$\longrightarrow \frac{ \sqrt{2} (5 \sqrt{2} - 5)}{6( \sqrt{3} + \sqrt{2} )}$

$\longrightarrow \frac{ 10 - 5 \sqrt{2} }{6( \sqrt{3} + \sqrt{2} )}$

$\longrightarrow \frac{ 10 - 5 \sqrt{2} }{6( \sqrt{3} + \sqrt{2} )} \times \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} - \sqrt{2} }$

$\longrightarrow \frac{ (10 - 5 \sqrt{2} )( \sqrt{3} - \sqrt{2}) }{6( 3 - 2)}$

$\longrightarrow \frac{ 10( \sqrt{3} - \sqrt{2} ) - 5 \sqrt{2}( \sqrt{3} - \sqrt{2} ) }{6}$

$\longrightarrow \frac{ 10\sqrt{3} - 10\sqrt{2} - 5 \sqrt{6} - 10}{6}$

$\longrightarrow \frac{ 5(2\sqrt{3} - 2\sqrt{2} - \sqrt{6} - 2)}{6}$

$\longrightarrow \frac{ 5}{6}(2\sqrt{3} - 2\sqrt{2} - \sqrt{6} - 2)$