Question

Find the value of :
$\dfrac{1}{ \sin(10) } - \dfrac{ \sqrt{3} }{ \cos(10) }$
​

Answer 1

Step-by-step explanation:

function y=cos−1(cosx) .

cos−1(cosx)=−x;−π≤x≤0

=x;0≤x≤π

=2π−x;π≤x≤2π

Answer 2

Question:

• Find the value of the expression  $\dfrac{1}{\sin(10)} - \dfrac{\sqrt{3} }{\cos(10)}$  

Answer:

• The value of the above given expression is 4

Step-by-step Explanation:

Given that:

• Trignometric expression $\tt \dfrac{1}{\sin(10)} - \dfrac{\sqrt{3} }{\cos(10)}$

To Find:

• The value of it after evaluating

Required Solution:

• Simplifying the expression

$\\ \\ {\purple{\bigstar \; {\bf{\underline{Subtracting \; the \; expressions :}} }}}$

${ : \implies } \tt \dfrac{1}{\sin(10)} - \dfrac{\sqrt{3} }{\cos(10)} \\ \\ \\ { : \implies } \tt \dfrac{\cos(10) - \sqrt{3} \sin(10)}{\sin(10) \;. \cos(10)}$

$\\ \\ {\purple{\bigstar \; {\bf{\underline{Multiplying \; with \; 1/2 \; on \; both \; sides : }}}}}$

${ : \implies } \tt \dfrac{\cos(10) - \sqrt{3} \sin(10)}{\sin(10) \;. \cos(10)} \\ \\ \\ { : \implies } \tt \dfrac{\dfrac{1}{2} \; . \;\cos(10) -\dfrac{ \sqrt{3}}{2} \sin(10)}{\dfrac{1}{2} \; . \;\sin(10) \;. \cos(10)}$

$\\ \\ {\purple{\bigstar \; {\bf{\underline{Rewriting \; the \; Expression \; we \; get :}}}}}$

${ : \implies } \tt \dfrac{\sin(30)\; . \;\cos(10) -\dfrac{ \sqrt{3}}{2} \sin(10)}{\dfrac{1}{2} \; . \;\sin(10) \;. \cos(10)} \qquad \bigg\lgroup \bf \sin(30) = \dfrac{1}{2} \bigg\rgroup\\ \\ \\ { : \implies } \tt \dfrac{\sin(30)\; . \;\cos(10) - \cos(30) \sin(10)}{\dfrac{1}{2} \; . \;\sin(10) \;. \cos(10)} \qquad \bigg\lgroup \bf \cos(30) = \dfrac{\sqrt{3}}{2} \bigg\rgroup$

$\\ \\ {\purple{\bigstar \; {\bf{\underline{Simplifying\; the \; Expression \; we \; get : }}}}}$

${ : \implies } \tt \dfrac{\sin(30)\; . \;\cos(10) - \cos(30) \sin(10)}{\dfrac{1}{2} \; . \;\sin(10) \;. \cos(10)} \\ \\ \\ { : \implies } \tt \dfrac{2 \times \sin(30)\; . \;\cos(10) - 2\times\cos(30) \sin(10)}{\;\sin(10) \;. \cos(10)}$

$\\ \\ {\purple{\bigstar \; {\bf{\underline{Multiplying \; with \; 2 \; on \; both \; Sides : }}}}}$

${ : \implies } \tt \dfrac{2 \times \sin(30)\; . \;\cos(10) - 2\times\cos(30) \sin(10)}{\;\sin(10) \;. \cos(10)} \\ \\ \\ { : \implies } \tt \dfrac{2 \times 2 \times \sin(30)\; . \;\cos(10) - 2\times\cos(30) \sin(10)}{2 \times \;\sin(10) \;. \cos(10)}$

$\\ \\ {\purple{\bigstar \; {\bf{\underline{Using \; trignometric \; Identities : }}}} }$

${ : \implies } \tt \dfrac{2 \times 2 \times \sin(30)\; . \;\cos(10) - 2\times\cos(30) \sin(10)}{2 \times \;\sin(10) \;. \cos(10)}\\ \\ \\ { : \implies } \tt \dfrac{4 \;\sin ( 30 - 10 )}{2 \times \;\sin(10) \;. \cos(10) )} \qquad \bigg\lgroup \sf \sin A \cos B - \cos B \sin A = \sin ( A - B ) \bigg \rgroup \\ \\ \\ { : \implies } \tt \dfrac{4 \;\sin ( 20 )}{\sin( 2\times 10 )} \qquad \bigg\lgroup \sf 2 \sin A \cos A = \sin 2A \bigg \rgroup$

$\\ \\ {\purple{\bigstar \; {\bf{\underline{On \; Further \; simplification \; we \; get : }}}}}$

${ : \implies } \tt \dfrac{4 \sin 20}{\sin( \times 10 )} \\ \\ \\ { : \implies } \tt \dfrac{4 \cancel{( \sin 20 )}}{\cancel{\sin20} } \\ \\ \\ { : \implies } {\pink{\underline{\boxed{\tt{ 4 }}}\bigstar}} \; \; \; \; \;\;\;\;$

Therefore:

• The value of the expression given above is 4

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