Question

The value of \[\sqrt {{\mathbf{3}}} {\mathbf{cosec2}}{{\mathbf{0}}^ \circ } - {\mathbf{sec2}}{{\mathbf{0}}^ \circ }\;\] is
A.2
B.$\dfrac{{2\sin 20^\circ }}{{\sin 40^\circ }}$
C.4
D.$\dfrac{{4\sin 20^\circ }}{{\sin 40^\circ }}$

Answer 1

Hint: To find the value of this trigonometric expression we will first convert the given trigonometric ratios into sin and cos. Then take LCM and solve. We will get the result in the form of \[sin\left( {A - B} \right)\] form, simplify that such that we will get the value of the required expression.

Complete step-by-step answer:
Given expression is \[\sqrt {{\mathbf{3}}} {\mathbf{cosec2}}{{\mathbf{0}}^ \circ } - {\mathbf{sec2}}{{\mathbf{0}}^ \circ }\;\]
Converting cosec and sec into sin and cos
We get,
\[ = \dfrac{{\sqrt {3 }}}{{sin{20}^\circ}} - \dfrac{1}{{cos{20}^\circ }}\]
Now multiplied and divide the numerator and denominator by 2 we get,
\[ = 2\left( {\dfrac{{\sqrt 3 /2}}{{sin{{20}^ \circ }}} - \dfrac{{1/2}}{{cos{{20}^ \circ }}}} \right)\]
Now write $\sin 60^\circ = \dfrac{{\sqrt 3 }}{2}$ and $\cos 60^\circ = \dfrac{1}{2}$
We get,
\[\Rightarrow 2\left( {\dfrac{{sin{{60}^\circ }}}{{sin{{20}^\circ }}} - \dfrac{{cos{{60}^\circ }}}{{cos{{20}^\circ }}}} \right)\]
Taking LCM and solving we get,
\[\Rightarrow 2\left( {\dfrac{{sin{{60}^ \circ }cos{{20}^ \circ } - cos{{60}^ \circ }sin{{20}^ \circ }}}{{sin{{20}^ \circ }cos{{20}^ \circ }}}} \right)\]
Applying sin (A-B) formula we get,
\[\Rightarrow 2\left( {\dfrac{{sin({{60}^ \circ } - {{20}^ \circ })}}{{sin{{20}^ \circ }cos{{20}^ \circ }}}} \right)\]
Or
\[\Rightarrow 2\left( {\dfrac{{sin{{40}^ \circ }}}{{sin{{20}^ \circ }cos{{20}^ \circ }}}} \right)\]
Now applying \[sin2A = 2sinAcosA\] formula
\[ = 2 \times \dfrac{{2sin{{20}^ \circ }cos{{20}^ \circ }}}{{sin{{20}^ \circ }cos{{20}^ \circ }}}\]
By cancelling the like terms
$ = 4$
Hence the value of the expression \[\sqrt {{\mathbf{3}}} {\mathbf{cosec2}}{{\mathbf{0}}^ \circ } - {\mathbf{sec2}}{{\mathbf{0}}^ \circ }\;\] is 4
So, the correct answer is “Option C”.

Note: Generally to solve this type of problems what we do is just convert first the given trigonometric expression into sin and cos trigonometric functions then either simplify or use formulas.