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[tex]\underline{\underline{\large{\red{\mathcal{✒GIVEN:}}}}}[/tex]
[tex]\bullet \: \: \rm{ \frac{5 { \sin}^{2} {30}^{ \circ} + { \cos}^{2}{45}^{ \circ} + 4 { \tan}^{2} {60}^{ \circ} }{2 \sin {30}^{ \circ} \cos {45}^{ \circ} + \tan {45}^{ \circ} }}[/tex]
[tex]\underline{\underline{\large{\red{\mathcal{REQUIRED:}}}}}[/tex]
Find the exact value.
[tex]\underline{\underline{\large{\red{\mathcal{SOLUTION:}}}}}[/tex]
Remember the six trigonometric ratios for [tex]\tt{\purple{special \: angles}}[/tex] [tex]\tt{{45}^{ \circ} , {30}^{ \circ} \: and \: {60}^{ \circ}}[/tex]:
[tex]\small{\boxed{ \bm{{ \red{ \sin {30}^{ \circ} = \dfrac{1}{2} }}}}}[/tex]
[tex]\small{\boxed{ \bm{{ \red{ \cos {45}^{ \circ} = \dfrac{ \sqrt{2} }{2} }}}}}[/tex]
[tex]\small{\boxed{ \bm{{ \red{ \tan {60}^{ \circ} = \sqrt{3} }}}}}[/tex]
[tex]\small{\boxed{ \bm{{ \red{ \tan {45}^{ \circ} = \sqrt{1} }}}}}[/tex]
Now, we substitute those values:
[tex]\small{\tt{ \frac{5( \frac{1}{2} {)}^{2} + ( \frac{ \sqrt{2} }{2} {)}^{2} + 4( \sqrt{3} {)}^{2} }{2 (\frac{1}{2}) ( \frac{ \sqrt{2} }{2} ) + 1} = \frac{ \frac{5}{4} + \frac{2}{4} + 12 }{ \frac{ \sqrt{2} }{2} + 1} }}[/tex]
[tex]\tt{ \frac{ \frac{7}{4} + 12}{ \frac{ \sqrt{2} + 2}{2} } = \frac{ \frac{7 + 48}{4} }{ \frac{ \sqrt{2} + 2}{2} } \div \frac{ \sqrt{2} + 2}{2} }[/tex]
[tex]\tt{ \dfrac{55(2)}{4( \sqrt{2} + 2)}}[/tex]
Simplify:
[tex]\large{\tt{\purple{ \dfrac{55}{2 \sqrt{2} + 4} }}}[/tex]
[tex]\tt{\therefore}[/tex] The exact value is [tex]\large{\rm{\purple{ \dfrac{55}{2 \sqrt{2} + 4} }}}[/tex].