Question

Evaluate the value of the following:
\[\dfrac{{\sin {{18}^0}}}{{\cos {{72}^0}}} + \sqrt 3 \left( {\tan {{10}^0}\tan {{40}^0}\tan {{30}^0}\tan {{50}^0}\tan {{80}^0}} \right)\].

Answer 1

Hint: The angles given here are not standard angles whose values you already know. First try to convert these angles into standard angles by subtracting or adding some angle from \[{90^0}\]. Then simplify the equation given and then proceed.

Complete step by step answer:
Let \[S = \dfrac{{\sin {{18}^0}}}{{\cos {{72}^0}}} + \sqrt 3 \left( {\tan {{10}^0}\tan {{40}^0}\tan {{30}^0}\tan {{50}^0}\tan {{80}^0}} \right)\]
We know that \[\tan {50^0} = \cot \left( {{{90}^0} - {{50}^0}} \right)\], \[\cos {72^0} = \sin \left( {{{90}^0} - {{72}^0}} \right)\] and \[\tan {80^0} = \cot \left( {{{90}^0} - {{80}^0}} \right)\]By using these formulae, we can rewrite the equation as
\[
   \Rightarrow S = \dfrac{{\sin {{18}^0}}}{{\sin \left( {{{90}^0} - {{72}^0}} \right)}} + \sqrt 3 \left( {\tan {{10}^0}\tan {{40}^0}\tan {{30}^0}\cot \left( {{{90}^0} - {{50}^0}} \right)\cot \left( {{{90}^0} - {{80}^0}} \right)} \right) \\
   \Rightarrow S = \dfrac{{\sin {{18}^0}}}{{\sin {{18}^0}}} + \sqrt 3 \left( {\tan {{10}^0}\tan {{40}^0}\tan {{30}^0}\cot {{40}^0}\cot {{10}^0}} \right) \\
\]
We know that, \[\cot {40^0} = \dfrac{1}{{\tan {{40}^0}}}\] and \[\cot {10^0} = \dfrac{1}{{\tan {{10}^0}}}\]. By substituting these values, we get
\[ \Rightarrow S = \dfrac{{\sin {{18}^0}}}{{\sin {{18}^0}}} + \sqrt 3 \left( {\tan {{10}^0}\tan {{40}^0}\tan {{30}^0}\dfrac{1}{{\cot {{40}^0}}}\dfrac{1}{{\cot {{10}^0}}}} \right)\]
Cancelling the common terms, we have
\[
   \Rightarrow S = \dfrac{{\sin {{18}^0}}}{{\sin {{18}^0}}} + \sqrt 3 \left( {\tan {{30}^0}} \right) \\
   \Rightarrow S = 1 + \sqrt 3 \left( {\dfrac{1}{{\sqrt 3 }}} \right){\text{ }}\left[ {\because \tan {{30}^0} = \dfrac{1}{{\sqrt 3 }}} \right] \\
   \Rightarrow S = 1 + 1 \\
  \therefore \dfrac{{\sin {{18}^0}}}{{\cos {{72}^0}}} + \sqrt 3 \left( {\tan {{10}^0}\tan {{40}^0}\tan {{30}^0}\tan {{50}^0}\tan {{80}^0}} \right) = 2 \\
\]

Thus, the value of \[\dfrac{{\sin {{18}^0}}}{{\cos {{72}^0}}} + \sqrt 3 \left( {\tan {{10}^0}\tan {{40}^0}\tan {{30}^0}\tan {{50}^0}\tan {{80}^0}} \right)\] is equal to 2.

Note: To solve these types of questions, use the formulae and conversions of trigonometric ratios except the known standard values. Try to cancel the terms so that we can reach our solution easily. Always remember the formulae in trigonometry to solve these kinds of problems.