Question

The value of expression \[\left( {\dfrac{1}{{cos{\text{ }}80^\circ }}} \right){\text{ }} - {\text{ }}\left( {\dfrac{{\sqrt 3 }}{{sin{\text{ }}80^\circ }}} \right)\] is equal to
\[1){\text{ }}\sqrt 2 \]
\[2){\text{ }}\sqrt 3 \]
\[3){\text{ }}2\]
\[4){\text{ }}4\]
\[5){\text{ }}\sqrt 5 \]

Answer 1

Hint: We have to find the value of the given trigonometric expression\[\left( {\dfrac{1}{{cos{\text{ }}80^\circ }}} \right){\text{ }} - {\text{ }}\left( {\dfrac{{\sqrt 3 }}{{sin{\text{ }}80^\circ }}} \right)\]. We solve this using the formula of double angle of sin function . We should also have the knowledge of values for various angles of trigonometric functions . Firstly we find the value of the given trigonometric expression using the formula of sum of two angles of a sine function . Also , the conversion of an angle in terms of another trigonometric function .

Complete step-by-step answer:
Given :
We have to find the value of the expression \[\left( {\dfrac{1}{{cos{\text{ }}80^\circ }}} \right){\text{ }} - {\text{ }}\left( {\dfrac{{\sqrt 3 }}{{sin{\text{ }}80^\circ }}} \right)\]
We know , that
\[cos{\text{ }}80^\circ {\text{ }} = {\text{ }}cos{\text{ }}\left( {90{\text{ }} - {\text{ }}10} \right)^\circ \]
The angle lies in the first quadrant and the value of the cosine function in the first quadrant is positive .
So ,
\[cos{\text{ }}80^\circ {\text{ }} = {\text{ }}sin{\text{ }}10^\circ \]
Similarly ,
\[sin{\text{ }}80^\circ {\text{ }} = {\text{ }}sin{\text{ }}\left( {90{\text{ }} - {\text{ }}10} \right)^\circ \]
The angle lies in the first quadrant and the value of the sine function in the first quadrant is positive .
So , \[sin{\text{ }}80^\circ = {\text{ }}cos{\text{ }}10^\circ \]
Now , the expression becomes
\[\left( {\dfrac{1}{{cos{\text{ }}80^\circ }}} \right){\text{ }} - {\text{ }}\left( {\dfrac{{\sqrt 3 }}{{sin{\text{ }}80^\circ }}} \right)\]\[ = {\text{ }}\left( {\dfrac{1}{{sin{\text{ }}10^\circ }}} \right){\text{ }} - {\text{ }}\left( {\dfrac{{\sqrt 3 }}{{cos{\text{ }}10^\circ }}} \right)\]
Taking L.C.M. we get
\[\left( {\dfrac{1}{{cos{\text{ }}80^\circ }}} \right){\text{ }} - {\text{ }}\left( {\dfrac{{\sqrt 3 }}{{sin{\text{ }}80^\circ }}} \right)\]\[ = {\text{ }}\dfrac{{\left[ {{\text{ }}cos{\text{ }}10^\circ {\text{ }} - {\text{ }}\sqrt 3 {\text{ }} \times {\text{ }}sin{\text{ }}10^\circ {\text{ }}} \right]}}{{\left[ {{\text{ }}sin{\text{ }}10^\circ {\text{ }} \times {\text{ }}cos{\text{ }}10^\circ {\text{ }}} \right]}}\]
Multiplying numerator and denominator by\[4\], we get
\[\left( {\dfrac{1}{{cos{\text{ }}80^\circ }}} \right){\text{ }} - {\text{ }}\left( {\dfrac{{\sqrt 3 }}{{sin{\text{ }}80^\circ }}} \right)\]\[ = {\text{ }}\dfrac{{4{\text{ }} \times {\text{ }}\left[ {{\text{ }}cos{\text{ }}10^\circ {\text{ }} - {\text{ }}\sqrt {3 \times } {\text{ }}sin{\text{ }}10^\circ {\text{ }}} \right]}}{{4{\text{ }} \times {\text{ }}\left[ {{\text{ }}sin{\text{ }}10^\circ {\text{ }} \times {\text{ }}cos{\text{ }}10^\circ {\text{ }}} \right]}}\]
Simplifying , we get
\[\left( {\dfrac{1}{{cos{\text{ }}80^\circ }}} \right){\text{ }} - {\text{ }}\left( {\dfrac{{\sqrt 3 }}{{sin{\text{ }}80^\circ }}} \right)\]\[ = {\text{ }}\dfrac{{\left[ {{\text{ }}\left( {\dfrac{1}{2}} \right){\text{ }} \times {\text{ }}cos{\text{ }}10^\circ {\text{ }} - {\text{ }}\left( {\dfrac{{\sqrt 3 }}{2}} \right){\text{ }} \times {\text{ }}sin{\text{ }}10^\circ {\text{ }}} \right]}}{{\left( {\dfrac{1}{4}} \right){\text{ }} \times \left[ {2{\text{ }} \times {\text{ }}sin{\text{ }}10^\circ {\text{ }} \times {\text{ }}cos{\text{ }}10^\circ {\text{ }}} \right]}}\]
We know that \[sin{\text{ }}2x{\text{ }} = {\text{ }}2{\text{ }}sin{\text{ }}x{\text{ }} \times {\text{ }}cos{\text{ }}x\]
We also know that
\[cos{\text{ }}30^\circ {\text{ }} = {\text{ }}\dfrac{{\sqrt 3 }}{2}\]
\[sin{\text{ }}30^\circ {\text{ }} = {\text{ }}\dfrac{1}{2}\]
Using these values , we get
\[\left( {\dfrac{1}{{cos{\text{ }}80^\circ }}} \right){\text{ }} - {\text{ }}\left( {\dfrac{{\sqrt 3 }}{{sin{\text{ }}80^\circ }}} \right)\]\[ = {\text{ }}\dfrac{{\left[ {{\text{ }}sin{\text{ }}30^\circ {\text{ }} \times {\text{ }}cos{\text{ }}10^\circ {\text{ }} - {\text{ }}cos{\text{ }}30^\circ {\text{ }} \times {\text{ }}sin{\text{ }}10^\circ {\text{ }}} \right]}}{{\left[ {{\text{ }}\left( {\dfrac{1}{4}} \right){\text{ }} \times {\text{ }}sin{\text{ }}20^\circ {\text{ }}} \right]}}\]
Also , the formula of difference of sin function is given as :
\[sin\left( {A{\text{ }} - {\text{ }}B} \right){\text{ }} = {\text{ }}sin{\text{ }}A{\text{ }} \times {\text{ }}cos{\text{ }}B{\text{ }} - {\text{ }}sin{\text{ }}B{\text{ }} \times {\text{ }}cos{\text{ }}A\]
Using this trigonometric formulas , we get
\[\left( {\dfrac{1}{{cos{\text{ }}80^\circ }}} \right){\text{ }} - {\text{ }}\left( {\dfrac{{\sqrt 3 }}{{sin{\text{ }}80^\circ }}} \right)\]\[ = \dfrac{{sin{\text{ }}20^\circ }}{{\left[ {{\text{ }}\left( {\dfrac{1}{4}} \right){\text{ }} \times {\text{ }}sin{\text{ }}20^\circ {\text{ }}} \right]}}\]
Cancelling the terms , we get
\[\left( {\dfrac{1}{{cos{\text{ }}80^\circ }}} \right){\text{ }} - {\text{ }}\left( {\dfrac{{\sqrt 3 }}{{sin{\text{ }}80^\circ }}} \right)\]\[ = {\text{ }}4\]
Thus the value of the expression is \[4\] .
Hence , the correct option is\[\left( 4 \right)\].
So, the correct answer is “Option 4”.

Note: The various trigonometric formulas we need to remember when by dividing or multiplying the given apply we are able to get a known trigonometric value are given as :
$sin{\text{ }}2x{\text{ }} = {\text{ }}2 \times {\text{ }}sin{\text{ }}x{\text{ }} \times {\text{ }}cos{\text{ }}x$
$cos2x = 2 \times {cos^2}x - 1 = 1 - 2 \times {sin^2}x$
$tan2x = \dfrac{{tan2x}}{{(1 - {tan^2}x)}}$
$sin3x = 3\sin x - 4{sin^3}x$
$cos3x = 4{cos^3}x - 3\cos x$