Answer

Verified

354.3k+ views

**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.

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

10 examples of law on inertia in our daily life

Write a letter to the principal requesting him to grant class 10 english CBSE

In 1946 the Interim Government was formed under a Sardar class 11 sst CBSE

Change the following sentences into negative and interrogative class 10 english CBSE