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# Evaluate: $\dfrac{{\sin 30 + \cos 30}}{{\sec 30 + \cos 30}}$

Last updated date: 16th Sep 2024
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Hint- $\sec \theta = \dfrac{1}{{\cos \theta }}$, and $\sin 30 = \dfrac{1}{2}$, $\cos 30 = \dfrac{{\sqrt 3 }}{2}$
As we know $\sin 30 = \dfrac{1}{2},{\text{ }}\cos 30 = \dfrac{{\sqrt 3 }}{2},{\text{ }}\sec 30 = \dfrac{1}{{\cos 30}} = \dfrac{2}{{\sqrt 3 }}$
$\Rightarrow \dfrac{{\sin 30 + \cos 30}}{{\sec 30 + \cos 30}} = \dfrac{{\dfrac{1}{2} + \dfrac{{\sqrt 3 }}{2}}}{{\dfrac{2}{{\sqrt 3 }} + \dfrac{{\sqrt 3 }}{2}}} = \dfrac{{\dfrac{{1 + \sqrt 3 }}{2}}}{{\dfrac{{2 \times 2 + \sqrt 3 \times \sqrt 3 }}{{2\sqrt 3 }}}} \\ = \dfrac{{1 + \sqrt 3 }}{2} \times \dfrac{{2\sqrt 3 }}{{\left( {4 + 3} \right)}} = \dfrac{{2\sqrt 3 + \left( {\sqrt 3 \times 2\sqrt 3 } \right)}}{{2 \times 7}} = \dfrac{{6 + 2\sqrt 3 }}{{14}} \\$
$\Rightarrow \dfrac{{\sin 30 + \cos 30}}{{\sec 30 + \cos 30}} = \dfrac{{3 + \sqrt 3 }}{7}$