Question

Evaluate the following expression.
$\dfrac{{\cos {{58}^ \circ }}}{{\sin {{32}^ \circ }}} + \dfrac{{\sin {{22}^ \circ }}}{{\cos {{68}^ \circ }}} - \dfrac{{\cos {{38}^ \circ }\cos ec{{52}^ \circ }}}{{\sqrt 3 \left( {\tan {{18}^ \circ }\tan {{35}^ \circ }\tan {{60}^ \circ }\tan {{72}^ \circ }\tan {{55}^ \circ }} \right)}}$

Answer 1

Hint : We know that, the value of $\cos A$ is equal to $\sin \left( {{{90}^ \circ } - A} \right)$, the value of $\cos ecA = \dfrac{1}{{\sin A}}$, value of $\tan A = \cot \left( {{{90}^ \circ } - A} \right)$, the value of $\tan A = \dfrac{1}{{\cot A}}$ and the value of $\sin A = \cos \left( {{{90}^ \circ } - A} \right)$. Use these trigonometric formulas to evaluate the given expression.

Complete step-by-step answer :
The angles of sine, cosine and tangent are the main functions of trigonometry. Other trigonometric functions are derived from these three functions itself.
So we are given to evaluate an expression.
The expression has 3 terms, first let’s consider the first two terms.
Expression with first two terms is $\dfrac{{\cos {{58}^ \circ }}}{{\sin {{32}^ \circ }}} + \dfrac{{\sin {{22}^ \circ }}}{{\cos {{68}^ \circ }}}$
The value of $\cos {58^ \circ }$ is equal to $\sin \left( {{{90}^ \circ } - {{58}^ \circ }} \right) = \sin {32^ \circ }$
The value of $\sin {22^ \circ }$ is equal to $\cos \left( {{{90}^ \circ } - {{22}^ \circ }} \right) = \cos {68^ \circ }$
Substitute the above obtained values in the two termed expression
$\dfrac{{\cos {{58}^ \circ }}}{{\sin {{32}^ \circ }}} + \dfrac{{\sin {{22}^ \circ }}}{{\cos {{68}^ \circ }}} \\\ \cos {58^ \circ } = \sin {32^ \circ },\sin {22^ \circ } = \cos {68^ \circ } \\\ \Rightarrow \dfrac{{\sin {{32}^ \circ }}}{{\sin {{32}^ \circ }}} + \dfrac{{\cos {{68}^ \circ }}}{{\cos {{68}^ \circ }}} \\\ \Rightarrow 1 + 1 = 2 \\\$
Now, evaluate the 3rd term $\dfrac{{\cos {{38}^ \circ }\cos ec{{52}^ \circ }}}{{\sqrt 3 \left( {\tan {{18}^ \circ }\tan {{35}^ \circ }\tan {{60}^ \circ }\tan {{72}^ \circ }\tan {{55}^ \circ }} \right)}}$
The value of $\cos ec{52^ \circ }$ is $\dfrac{1}{{\sin {{52}^ \circ }}}$
The value of $\sin {52^ \circ } = \cos \left( {{{90}^ \circ } - {{52}^ \circ }} \right) = \cos {38^ \circ }$
The value of $\tan {18^ \circ }$ is equal to $\cot \left( {{{90}^ \circ } - {{18}^ \circ }} \right) = \cot {72^ \circ }$
The value of $\tan {35^ \circ }$ is equal to $\cot \left( {{{90}^ \circ } - {{35}^ \circ }} \right) = \cot {55^ \circ }$
The value of $\cot {72^ \circ }$ is equal to $\dfrac{1}{{\tan {{72}^ \circ }}}$
The value of $\cot {55^ \circ }$ is equal to $\dfrac{1}{{\tan {{55}^ \circ }}}$
On substituting the above obtained values in the 3rd term of the expression, we get
$\dfrac{{\cos {{38}^ \circ }\cos ec{{52}^ \circ }}}{{\sqrt 3 \left( {\tan {{18}^ \circ }\tan {{35}^ \circ }\tan {{60}^ \circ }\tan {{72}^ \circ }\tan {{55}^ \circ }} \right)}} \\\ \cos ec{52^ \circ } = \dfrac{1}{{\sin {{52}^ \circ }}},\tan {18^ \circ } = \cot {72^ \circ },\tan {35^ \circ } = \cot {55^ \circ } \\\ = \dfrac{{\cos {{38}^ \circ }\left( {\dfrac{1}{{\sin {{52}^ \circ }}}} \right)}}{{\sqrt 3 \left( {\cot {{72}^ \circ }\cot {{55}^ \circ }\tan {{60}^ \circ }\tan {{72}^ \circ }\tan {{55}^ \circ }} \right)}} \\\ \sin {52^ \circ } = \cos {38^ \circ },\cot {72^ \circ } = \dfrac{1}{{\tan {{72}^ \circ }}},\cot {55^ \circ } = \dfrac{1}{{\tan {{55}^ \circ }}} \\\ = \dfrac{{\cos {{38}^ \circ }\left( {\dfrac{1}{{\cos {{38}^ \circ }}}} \right)}}{{\sqrt 3 \left( {\dfrac{1}{{\tan {{72}^ \circ }}} \times \dfrac{1}{{\tan {{55}^ \circ }}}\tan {{60}^ \circ }\tan {{72}^ \circ }\tan {{55}^ \circ }} \right)}} \\\ = \dfrac{1}{{\sqrt 3 \left( {1 \times 1 \times \tan {{60}^ \circ }} \right)}} \\\$
The value of $\tan {60^ \circ }$ is equal to $\sqrt 3$
On substituting the value of $\tan {60^ \circ }$ in the above expression, we get
$\dfrac{1}{{\sqrt 3 \left( {\tan {{60}^ \circ }} \right)}} = \dfrac{1}{{\sqrt 3 \times \sqrt 3 }} = \dfrac{1}{3}$
Now substitute the values of the terms in the main expression
$\dfrac{{\cos {{58}^ \circ }}}{{\sin {{32}^ \circ }}} + \dfrac{{\sin {{22}^ \circ }}}{{\cos {{68}^ \circ }}} - \dfrac{{\cos {{38}^ \circ }\cos ec{{52}^ \circ }}}{{\sqrt 3 \left( {\tan {{18}^ \circ }\tan {{35}^ \circ }\tan {{60}^ \circ }\tan {{72}^ \circ }\tan {{55}^ \circ }} \right)}} \\\ = 1 + 1 - \dfrac{1}{3} \\\ = 2 - \dfrac{1}{3} \\\ = \dfrac{5}{3} \\\$
Therefore, the value of the given expression is $\dfrac{5}{3}$

Note : The value of trigonometric functions can be obtained from a right angled triangle, where one angle is 90 degrees, and if one of the other two angles is known then the other angle can be obtained.
The value of sine of an angle is the ratio of the side opposite to angle and hypotenuse.
The value of cosine of an angle is the ratio of the side adjacent to angle and hypotenuse.
The value of tangent of an angle is the ratio of side opposite to angle and side adjacent to angle, and also it is the ratio of sine function to cosine function.