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

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Last updated date: 29th May 2024
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Answer
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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 $\sin 30 = \dfrac{1}{2},{\text{ }}\cos 30 = \dfrac{{\sqrt 3 }}{2},{\text{ }}\sec 30 = \dfrac{1}{{\cos 30}} = \dfrac{2}{{\sqrt 3 }}$
So, substitute these values in the given equation
$
   \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}} \\
$
Now divide by 2 in numerator and denominator
$ \Rightarrow \dfrac{{\sin 30 + \cos 30}}{{\sec 30 + \cos 30}} = \dfrac{{3 + \sqrt 3 }}{7}$
So, this is the required answer.

Note: - In such types of questions the key concept is that we have to remember all the standard angle values, then substitute these values in the given equation then simplify it we will get the required answer.

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