Answer:
[tex]\boxed{\bf\:5 \bigg( {x}^{2} - \dfrac{1}{ {x}^{2} } \bigg) = \dfrac{1}{5} \: } \\ [/tex]
Step-by-step explanation:
Given that,
[tex]\sf\: 5x = sec \theta \\ [/tex]
[tex]\sf\: \dfrac{5}{x} = tan \theta \\ [/tex]
Now, We know
[tex]\sf\: {sec}^{2} \theta - {tan}^{2} \theta = 1 \\ [/tex]
On substituting the values from given, we get
[tex]\sf\: {(5x)}^{2} - {\bigg( \dfrac{5}{x} \bigg) }^{2} = 1 \\ [/tex]
[tex]\sf\: {25x}^{2} - \dfrac{25}{ {x}^{2} } = 1 \\ [/tex]
[tex]\sf\:25 \bigg( {x}^{2} - \dfrac{1}{ {x}^{2} } \bigg) = 1 \\ [/tex]
On dividing both sides by 5, we get
[tex]\implies\sf\:5 \bigg( {x}^{2} - \dfrac{1}{ {x}^{2} } \bigg) = \dfrac{1}{5} \\ [/tex]
Hence,
[tex]\implies\sf\:\boxed{\bf\:5 \bigg( {x}^{2} - \dfrac{1}{ {x}^{2} } \bigg) = \dfrac{1}{5} \: } \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Short-cut Trick:
[tex]\boxed{\sf\:\bf\:If \: sec\theta = ax, \: tan\theta = \dfrac{a}{x}, \: then \: a \bigg( {x}^{2} - \dfrac{1}{ {x}^{2} } \bigg) = \dfrac{1}{a} \: } \\ [/tex]
