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Evaluate the following.
1. \[\dfrac{{\cos {{45}^ \circ }}}{{\sec {{30}^ \circ } + \cos ec{{30}^ \circ }}}\]
2. \[\dfrac{{\sin {{30}^ \circ } + \tan {{45}^ \circ } - \cos ec{{60}^ \circ }}}{{\sec {{30}^ \circ } + \cos {{60}^ \circ } + \cot {{45}^ \circ }}}\]
3. \[\dfrac{{5{{\cos }^2}{{60}^ \circ } + 4{{\sec }^2}{{30}^ \circ } - {{\tan }^2}{{45}^ \circ }}}{{{{\sin }^2}{{30}^ \circ } + {{\cos }^2}{{30}^ \circ }}}\]

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Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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Hint: To evaluate the trigonometric ratios above we just need the values of these functions for respective angles. Also we need to know the basic trigonometric identities. We will simply tabulate the values of functions and angles. Then we will substitute the values on their asked trigonometric identity. If required we will take LCM and simplify the ratio.

Complete Step by Step Solution:
We will write the basic values of the functions for the angles so taken.

FunctionAnglessincostancotsecCosec
\[{30^ \circ }\]\[\dfrac{1}{2}\]\[\dfrac{{\sqrt 3 }}{2}\]\[\dfrac{1}{{\sqrt 3 }}\]\[\sqrt 3 \]\[\dfrac{2}{{\sqrt 3 }}\]2
\[{45^ \circ }\]\[\dfrac{1}{{\sqrt 2 }}\]\[\dfrac{1}{{\sqrt 2 }}\]\[1\]1\[\sqrt 2 \]\[\sqrt 2 \]
\[{60^ \circ }\]\[\dfrac{{\sqrt 3 }}{2}\]\[\dfrac{1}{2}\]\[\sqrt 3 \]\[\dfrac{1}{{\sqrt 3 }}\]2\[\dfrac{2}{{\sqrt 3 }}\]

Now the table is completed we will directly use these values from the table and we will put them in the problems.
1. \[\dfrac{{\cos {{45}^ \circ }}}{{\sec {{30}^ \circ } + \cos ec{{30}^ \circ }}}\]
Putting the values we get,
\[ \Rightarrow \dfrac{{\dfrac{1}{{\sqrt 2 }}}}{{\dfrac{2}{{\sqrt 3 }} + 2}}\]
Taking LCM of denominator we get,
\[ \Rightarrow \dfrac{{\dfrac{1}{{\sqrt 2 }}}}{{\dfrac{{2 + 2\sqrt 3 }}{{\sqrt 3 }}}}\]
Now rearranging the terms we get,
\[ \Rightarrow \dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 3 }}{{2 + 2\sqrt 3 }}\]
on multiplying the denominator,
\[ \Rightarrow \dfrac{{\sqrt 3 }}{{2\sqrt 2 + 2\sqrt 5 }}\]
This is our answer.

2. \[\dfrac{{\sin {{30}^ \circ } + \tan {{45}^ \circ } - \cos ec{{60}^ \circ }}}{{\sec {{30}^ \circ } + \cos {{60}^ \circ } + \cot {{45}^ \circ }}}\]
Putting the values we get,
\[ \Rightarrow \dfrac{{\dfrac{1}{2} + 1 - \dfrac{2}{{\sqrt 3 }}}}{{2 + \dfrac{1}{2} + 1}}\]
Taking LCM of numerator and denominator
\[ \Rightarrow \dfrac{{\dfrac{{\sqrt 3 + 2\sqrt 3 - 2 \times 2}}{{2\sqrt 3 }}}}{{\dfrac{{2 \times 2 + 2 + 2}}{2}}}\]
On solving the terms
\[ \Rightarrow \dfrac{{\dfrac{{3\sqrt 3 - 4}}{{2\sqrt 3 }}}}{{\dfrac{8}{2}}}\]
Again simplifying the ratios
\[ \Rightarrow \dfrac{{3\sqrt 3 - 4}}{{2\sqrt 3 }} \times \dfrac{1}{4}\]
\[ \Rightarrow \dfrac{{3\sqrt 3 - 4}}{{8\sqrt 3 }}\]
This is our answer.

3. \[\dfrac{{5{{\cos }^2}{{60}^ \circ } + 4{{\sec }^2}{{30}^ \circ } - {{\tan }^2}{{45}^ \circ }}}{{{{\sin }^2}{{30}^ \circ } + {{\cos }^2}{{30}^ \circ }}}\]
This equation above can also be written as,
\[ \Rightarrow \dfrac{{5{{\left( {\cos {{60}^ \circ }} \right)}^2} + 4{{\left( {\sec {{30}^ \circ }} \right)}^2} - {{\left( {\tan {{45}^ \circ }} \right)}^2}}}{{{{\sin }^2}{{30}^ \circ } + {{\cos }^2}{{30}^ \circ }}}\]
In the denominator we know that \[{\sin ^2}\theta + {\cos ^2}\theta = 1\] . so denominator is 1. So we can rewrite the expression again as,
\[ \Rightarrow 5{\left( {\cos {{60}^ \circ }} \right)^2} + 4{\left( {\sec {{30}^ \circ }} \right)^2} - {\left( {\tan {{45}^ \circ }} \right)^2}\]
Putting the values we get,
\[ \Rightarrow 5{\left( {\dfrac{1}{2}} \right)^2} + 4{\left( {\dfrac{2}{{\sqrt 3 }}} \right)^2} - {\left( 1 \right)^2}\]
Taking the squares and solving the ratios,
\[ \Rightarrow \dfrac{5}{4} + \dfrac{{4 \times 4}}{3} - 1\]
Taking LCM
\[ \Rightarrow \dfrac{{5 \times 3 + 16 \times 4 - 4 \times 3}}{{12}}\]
And then solving again,
\[ \Rightarrow \dfrac{{15 + 64 - 12}}{{12}}\]
\[ \Rightarrow \dfrac{{67}}{{12}}\]
This is our answer. To further simplify we will get the answer in decimals.
So,
\[\dfrac{{\cos {{45}^ \circ }}}{{\sec {{30}^ \circ } + \cos ec{{30}^ \circ }}}\]\[ \Rightarrow \dfrac{{\sqrt 3 }}{{2\sqrt 2 + 2\sqrt 5 }}\]
\[\dfrac{{\sin {{30}^ \circ } + \tan {{45}^ \circ } - \cos ec{{60}^ \circ }}}{{\sec {{30}^ \circ } + \cos {{60}^ \circ } + \cot {{45}^ \circ }}}\]\[ \Rightarrow \dfrac{{3\sqrt 3 - 4}}{{8\sqrt 3 }}\]
\[\dfrac{{5{{\cos }^2}{{60}^ \circ } + 4{{\sec }^2}{{30}^ \circ } - {{\tan }^2}{{45}^ \circ }}}{{{{\sin }^2}{{30}^ \circ } + {{\cos }^2}{{30}^ \circ }}}\]\[ \Rightarrow \dfrac{{67}}{{12}}\]


Note:
Evaluating the question is nothing but finding the solution. Not only a solution but in simplest form as much as possible. Here we used some identities also along with the trigonometric functions, their angles and their values. Remember that the terms with the same root only can be added or simplified.
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