Answer
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Hint: The given question deals with basic simplification of trigonometric expression by using some of the simple trigonometric formulae, trigonometric identities and values of some trigonometric ratios for some basic and standard angles. Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem.
Complete step by step solution:
In the given problem, we have to find the value of trigonometric expression: $6736{\cos ^2}{18^ \circ } + 421{\tan ^2}{36^ \circ }$.
So, we know the values of the trigonometric function for the angles ${36^ \circ }$ and ${18^ \circ }$. Hence, we can substitute the values and simplify the expression further.
So, putting in the value of trigonometric function $\cos \left( {{{18}^ \circ }} \right)$ as $\left( {\dfrac{{\sqrt {10 + 2\sqrt 5 } }}{4}} \right)$ and the value of $\tan \left( {{{36}^ \circ }} \right)$ as $\left( {\dfrac{{\sqrt {10 - 2\sqrt 5 } }}{{\sqrt 5 + 1}}} \right)$. Hence, we get,
$ \Rightarrow 6736{\left( {\dfrac{{\sqrt {10 + 2\sqrt 5 } }}{4}} \right)^2} + 421{\left( {\dfrac{{\sqrt {10 - 2\sqrt 5 } }}{{\sqrt 5 + 1}}} \right)^2}$
Evaluating the squares of the terms and brackets, we get,
$ \Rightarrow 6736\left( {\dfrac{{10 + 2\sqrt 5 }}{{16}}} \right) + 421\left( {\dfrac{{10 - 2\sqrt 5 }}{{6 + 2\sqrt 5 }}} \right)$
Now, we have to simplify the above expression using the basic simplification rules.
Cancelling the common factors in numerator and denominator, we get,
\[ \Rightarrow 421\left( {10 + 2\sqrt 5 } \right) + 421\left( {\dfrac{{10 - 2\sqrt 5 }}{{6 + 2\sqrt 5 }}} \right)\]
Now, we have to rationalize the denominator of the second term. So, we get,
\[ \Rightarrow 421\left( {10 + 2\sqrt 5 } \right) + 421\left( {\dfrac{{10 - 2\sqrt 5 }}{{6 + 2\sqrt 5 }}} \right)\left( {\dfrac{{6 - 2\sqrt 5 }}{{6 - 2\sqrt 5 }}} \right)\]
Simplifying the expression further, we get,
\[ \Rightarrow 421\left( {10 + 2\sqrt 5 } \right) + 421\left( {\dfrac{{\left( {10 - 2\sqrt 5 } \right)\left( {6 - 2\sqrt 5 } \right)}}{{{{\left( 6 \right)}^2} - {{\left( {2\sqrt 5 } \right)}^2}}}} \right)\]
\[ \Rightarrow 421\left( {10 + 2\sqrt 5 } \right) + 421\left( {\dfrac{{60 - 12\sqrt 5 - 20\sqrt 5 + 20}}{{36 - 20}}} \right)\]
\[ \Rightarrow 421\left( {10 + 2\sqrt 5 } \right) + 421\left( {\dfrac{{80 - 32\sqrt 5 }}{{16}}} \right)\]
Cancelling the common factors in numerator and denominator, we get,
\[ \Rightarrow 421\left( {10 + 2\sqrt 5 } \right) + 421\left( {5 - 2\sqrt 5 } \right)\]
Now, taking $421$common from both the terms, we get,
\[ \Rightarrow 421\left[ {\left( {10 + 2\sqrt 5 } \right) + \left( {5 - 2\sqrt 5 } \right)} \right]\]
\[ \Rightarrow 421 \times 15\]
\[ \Rightarrow 6315\]
Hence, the value of $6736{\cos ^2}{18^ \circ } + 421{\tan ^2}{36^ \circ }$ is \[6315\] by the use of basic algebraic rules and simple trigonometric formulae and values.
So, the correct answer is “6315”.
Note: Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart. Besides these simple trigonometric formulae, trigonometric identities are also of significant use in such types of questions where we have to simplify trigonometric expressions with help of basic knowledge of algebraic rules and operations. One must know the values of trigonometric functions for the angles ${36^ \circ }$ and ${18^ \circ }$ in order to solve the problem.
Complete step by step solution:
In the given problem, we have to find the value of trigonometric expression: $6736{\cos ^2}{18^ \circ } + 421{\tan ^2}{36^ \circ }$.
So, we know the values of the trigonometric function for the angles ${36^ \circ }$ and ${18^ \circ }$. Hence, we can substitute the values and simplify the expression further.
So, putting in the value of trigonometric function $\cos \left( {{{18}^ \circ }} \right)$ as $\left( {\dfrac{{\sqrt {10 + 2\sqrt 5 } }}{4}} \right)$ and the value of $\tan \left( {{{36}^ \circ }} \right)$ as $\left( {\dfrac{{\sqrt {10 - 2\sqrt 5 } }}{{\sqrt 5 + 1}}} \right)$. Hence, we get,
$ \Rightarrow 6736{\left( {\dfrac{{\sqrt {10 + 2\sqrt 5 } }}{4}} \right)^2} + 421{\left( {\dfrac{{\sqrt {10 - 2\sqrt 5 } }}{{\sqrt 5 + 1}}} \right)^2}$
Evaluating the squares of the terms and brackets, we get,
$ \Rightarrow 6736\left( {\dfrac{{10 + 2\sqrt 5 }}{{16}}} \right) + 421\left( {\dfrac{{10 - 2\sqrt 5 }}{{6 + 2\sqrt 5 }}} \right)$
Now, we have to simplify the above expression using the basic simplification rules.
Cancelling the common factors in numerator and denominator, we get,
\[ \Rightarrow 421\left( {10 + 2\sqrt 5 } \right) + 421\left( {\dfrac{{10 - 2\sqrt 5 }}{{6 + 2\sqrt 5 }}} \right)\]
Now, we have to rationalize the denominator of the second term. So, we get,
\[ \Rightarrow 421\left( {10 + 2\sqrt 5 } \right) + 421\left( {\dfrac{{10 - 2\sqrt 5 }}{{6 + 2\sqrt 5 }}} \right)\left( {\dfrac{{6 - 2\sqrt 5 }}{{6 - 2\sqrt 5 }}} \right)\]
Simplifying the expression further, we get,
\[ \Rightarrow 421\left( {10 + 2\sqrt 5 } \right) + 421\left( {\dfrac{{\left( {10 - 2\sqrt 5 } \right)\left( {6 - 2\sqrt 5 } \right)}}{{{{\left( 6 \right)}^2} - {{\left( {2\sqrt 5 } \right)}^2}}}} \right)\]
\[ \Rightarrow 421\left( {10 + 2\sqrt 5 } \right) + 421\left( {\dfrac{{60 - 12\sqrt 5 - 20\sqrt 5 + 20}}{{36 - 20}}} \right)\]
\[ \Rightarrow 421\left( {10 + 2\sqrt 5 } \right) + 421\left( {\dfrac{{80 - 32\sqrt 5 }}{{16}}} \right)\]
Cancelling the common factors in numerator and denominator, we get,
\[ \Rightarrow 421\left( {10 + 2\sqrt 5 } \right) + 421\left( {5 - 2\sqrt 5 } \right)\]
Now, taking $421$common from both the terms, we get,
\[ \Rightarrow 421\left[ {\left( {10 + 2\sqrt 5 } \right) + \left( {5 - 2\sqrt 5 } \right)} \right]\]
\[ \Rightarrow 421 \times 15\]
\[ \Rightarrow 6315\]
Hence, the value of $6736{\cos ^2}{18^ \circ } + 421{\tan ^2}{36^ \circ }$ is \[6315\] by the use of basic algebraic rules and simple trigonometric formulae and values.
So, the correct answer is “6315”.
Note: Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart. Besides these simple trigonometric formulae, trigonometric identities are also of significant use in such types of questions where we have to simplify trigonometric expressions with help of basic knowledge of algebraic rules and operations. One must know the values of trigonometric functions for the angles ${36^ \circ }$ and ${18^ \circ }$ in order to solve the problem.