Question

Find:
4sin^60° - cos^45°
______________
tan^30° + sin^0°

Answer 1

$\red{ \huge{ \underline{ \overline{ \mid{ \bold{ \purple{ \: \: SOLUTION \: \: \red{ \mid}}}}}}}}$

$\underline{ \bold{ \: \: We \: \: know \: \: that \: \: }} \\ \\ \bold{sin \: 60 \degree \: = \frac{ \sqrt{3} }{2} } \\ \\ \bold{cos \: 45 \degree = \frac{1}{ \sqrt{2} } } \\ \\ \bold{tan \: 30 \degree = \frac{1}{ \sqrt{3} } } \\ \\ \bold{ sin \: 0 \degree = 0} \\ \\ \\ \\ \underline{ \bold{ \: \: Putting \: \: the \: values \: \: we \: \: get \: \: }}$

$\bold{ \frac{4 \: sin \: 60 \degree \: - \: cos \: 45 \degree}{tan \: 30 \degree \: + \: sin \: 0 \degree} } \\ \\ \bold{ = \frac{4 \times \frac{ \sqrt{3} }{2} \: - \: \frac{1}{ \sqrt{2} } }{ \frac{1}{ \sqrt{3} } + \: 0} } \\ \\ \bold{ = \frac{2 \sqrt{3} - \frac{1}{\sqrt{2} }}{ \frac{1}{ \sqrt{3} } } } \\ \\ \bold{ = \frac{ \frac{2 \sqrt{6} - 1}{\sqrt{2}} }{ \frac{1}{ \sqrt{3} } } } \\ \\ \bold{ = \frac{2\sqrt{6} - 1}{\sqrt{2}} \times \sqrt{3} } \\ \\ \bold{ = \frac{2 \sqrt{6} \times \sqrt{3} - \sqrt{3} }{\sqrt{2} }} \\ \\ \bold{ = \frac{2 \sqrt{18} - \sqrt{3} }{\sqrt{2} }} \\ \\ \bold{= \frac{6 \sqrt{2} - \sqrt{3}}{\sqrt{2}}} \\ \\ \bold{= \frac{\sqrt{2} ( 6 \sqrt{2} - \sqrt {3}) }{\sqrt{2} \times \sqrt{2}}} \\ \\ \bold{= \frac{ 12 - \sqrt{6}}{2}}$

Answer 2

$\huge\mathtt\red{Answer}$

$\begin{lgathered}\underline{ \bold{ \: \: We \: \: know \: \: that \: \: }} \\ \\ \bold{sin \: 60 \degree \: = \frac{ \sqrt{3} }{2} } \\ \\ \bold{cos \: 45 \degree = \frac{1}{ \sqrt{2} } } \\ \\ \bold{tan \: 30 \degree = \frac{1}{ \sqrt{3} } } \\ \\ \bold{ sin \: 0 \degree = 0} \\ \\ \\ \\ \underline{ \bold{ \: \: Putting \: \: the \: values \: \: we \: \: get \: \: }}\end{lgathered}$

$\begin{lgathered}\bold{ \frac{4 \: sin \: 60 \degree \: - \: cos \: 45 \degree}{tan \: 30 \degree \: + \: sin \: 0 \degree} } \\ \\ \bold{ = \frac{4 \times \frac{ \sqrt{3} }{2} \: - \: \frac{1}{ \sqrt{2} } }{ \frac{1}{ \sqrt{3} } + \: 0} } \\ \\ \bold{ = \frac{2 \sqrt{3} - \frac{1}{\sqrt{2} }}{ \frac{1}{ \sqrt{3} } } } \\ \\ \bold{ = \frac{ \frac{2 \sqrt{6} - 1}{\sqrt{2}} }{ \frac{1}{ \sqrt{3} } } } \\ \\ \bold{ = \frac{2\sqrt{6} - 1}{\sqrt{2}} \times \sqrt{3} } \\ \\ \bold{ = \frac{2 \sqrt{6} \times \sqrt{3} - \sqrt{3} }{\sqrt{2} }} \\ \\ \bold{ = \frac{2 \sqrt{18} - \sqrt{3} }{\sqrt{2} }} \\ \\ \bold{= \frac{6 \sqrt{2} - \sqrt{3}}{\sqrt{2}}} \\ \\ \bold{= \frac{\sqrt{2} ( 6 \sqrt{2} - \sqrt {3}) }{\sqrt{2} \times \sqrt{2}}} \\ \\ \bold{= \frac{ 12 - \sqrt{6}}{2}}\end{lgathered}$