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# Find the value of $\dfrac{{\cos {{45}^ \circ }}}{{\sec {{30}^ \circ } + {\text{cosec}}{{30}^ \circ }}}$.

Last updated date: 31st Mar 2023
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Hint: Put the value of $\cos {45^ \circ }$, $\sec {30^ \circ }$ and ${\text{cosec}}{30^ \circ }$ in the expression and find out its value.
According to the question, we have to calculate the value of $\dfrac{{\cos {{45}^ \circ }}}{{\sec {{30}^ \circ } + {\text{cosec}}{{30}^ \circ }}}$ .
We know that $\cos {45^ \circ } = \dfrac{1}{{\sqrt 2 }},\sec {30^ \circ } = \dfrac{2}{{\sqrt 3 }}$ and ${\text{cosec}}{30^ \circ } = 2$. So, putting all these values in the above expression, we’ll get:
$\Rightarrow \dfrac{{\cos {{45}^ \circ }}}{{\sec {{30}^ \circ } + {\text{cosec}}{{30}^ \circ }}} = \dfrac{{\dfrac{1}{{\sqrt 2 }}}}{{\dfrac{2}{{\sqrt 3 }} + 2}}, \\ \Rightarrow \dfrac{{\cos {{45}^ \circ }}}{{\sec {{30}^ \circ } + {\text{cosec}}{{30}^ \circ }}} = \dfrac{{\dfrac{1}{{\sqrt 2 }}}}{{\dfrac{{2 + 2\sqrt 3 }}{{\sqrt 3 }}}}, \\ \Rightarrow \dfrac{{\cos {{45}^ \circ }}}{{\sec {{30}^ \circ } + {\text{cosec}}{{30}^ \circ }}} = \dfrac{{\sqrt 3 }}{{\sqrt 2 \left( {2 + 2\sqrt 3 } \right)}} \\$
Therefore the value of $\dfrac{{\cos {{45}^ \circ }}}{{\sec {{30}^ \circ } + {\text{cosec}}{{30}^ \circ }}}$ is $\dfrac{{\sqrt 3 }}{{\sqrt 2 \left( {2 + 2\sqrt 3 } \right)}}$.