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# Solve the following trigonometric equation and find the value${\left( {\dfrac{{\sin {{47}^ \circ }}}{{\cos {{43}^ \circ }}}} \right)^2} + {\left( {\dfrac{{\cos{{43}^ \circ }}}{{\sin {{47}^ \circ }}}} \right)^2} - 4{\cos ^2}{45^ \circ }$

Last updated date: 20th Jun 2024
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Hint: - Try to break the angle as a sum of other angles with multiple of${90^ \circ },{180^ \circ },{270^ \circ }\& {360^ \circ }$.
We have to find the value of${\left( {\dfrac{{\sin {{47}^ \circ }}}{{\cos {{43}^ \circ }}}} \right)^2} + {\left( {\dfrac{{\cos {{43}^ \circ }}}{{\sin {{47}^ \circ }}}} \right)^2} - 4{\cos ^2}{45^ \circ }$
$\left[ {\sin \left( {{{90}^ \circ } - \theta } \right) = \cos \theta ,\cos \left( {{{90}^ \circ } - \theta } \right) = \sin \theta \& \cos {{45}^ \circ } = \dfrac{1}{{\sqrt 2 }}} \right]$
$\Rightarrow {\left( {\dfrac{{\sin \left( {{{90}^ \circ } - {{43}^ \circ }} \right)}}{{\cos {{43}^ \circ }}}} \right)^2} + {\left( {\dfrac{{\cos \left( {{{90}^ \circ } - {{47}^ \circ }} \right)}}{{\sin {{47}^ \circ }}}} \right)^2} - 4{\left( {\dfrac{1}{{\sqrt 2 }}} \right)^2} \\ \Rightarrow 1 + 1 - 4\left( {\dfrac{1}{2}} \right) \\ \Rightarrow 2 - 2 \\ \Rightarrow 0 \\$