Question

`\footnotesize \displaystyle\sf \prod^( \infty )_(n = 1) \bigg( (25)/(777) \bigg)^{ {( - 1)}^(n) \bigg( ( \tan^(4n) (x) + 1)/( \tan ^(2n) (x) ) \bigg) }`

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Answer 1

`\large\underline{\sf{Solution-}}`

Given expression is

`\rm :\longmapsto\:\prod \limits^( \infty )_(n = 1) \bigg( (25)/(777) \bigg)^{ {( - 1)}^(n) \bigg( ( tan^(4n) (x) + 1)/( tan ^(2n) (x) ) \bigg)}`

can be rewritten as

`\rm \: = \: \prod \limits^( \infty )_(n = 1) \bigg( (25)/(777) \bigg)^{ {( - 1)}^(n) \bigg( ( tan^(4n) (x))/( tan ^(2n) (x)) + \frac{1}{ {tan}^(2n) (x)} \bigg)}`

`\rm \: = \: \prod \limits^( \infty )_(n = 1) \bigg( (25)/(777) \bigg)^{ {( - 1)}^(n) \bigg( {tan}^(2n)(x) + {cot}^(2n)(x) \bigg)}`

can be further rewritten as

`\begin{gathered}\rm \: = \: \prod \limits^( \infty )_(n = 1) \bigg( (25)/(777) \bigg)^{ {( - 1)}^(n) \bigg( {( {tan}^(2)x) }^(n) + {( {cot}^(2) x)}^(n) \bigg)} \\ \end{gathered}`

`\begin{gathered}\rm \: = \: \prod \limits^( \infty )_(n = 1) \bigg( (25)/(777) \bigg)^{\bigg( { {( - 1)}^(n) ( {tan}^(2)x) }^(n) + {( - 1)}^(n) {( {cot}^(2) x)}^(n) \bigg)} \\ \end{gathered}`

`\begin{gathered}\rm \: = \: \prod \limits^( \infty )_(n = 1) \bigg( (25)/(777) \bigg)^{ \bigg( {( { - tan}^(2)x) }^(n) + {( { - cot}^(2) x)}^(n) \bigg)} \\ \end{gathered}`

`\rm \: = \: \bigg( (25)/(777) \bigg)^{ \bigg( - {tan}^(2)x - {cot}^(2)x + {tan}^(4)x + {cot}^(4)x - {tan}^(6)x - {cot}^(6)x + - - \bigg)}`

`\begin{gathered}\rm \: = \: \bigg( (25)/(777) \bigg)^{ \bigg(( - {tan}^(2)x + {tan}^(4)x - {tan}^(6)x + - - ) + ( - {cot}^(2)x + {cot}^(4)x - {cot}^(6)x + - -) \bigg)} \\ \end{gathered}`

Using Sum of infinite GP series,

`\begin{gathered} \purple{\rm :\longmapsto\:\boxed{\tt S_ \infty \: = \: (a)/(1 - r) \: where \: }} \\ \end{gathered}`

So, using this,

`\begin{gathered}\rm \: = \: \bigg( (25)/(777) \bigg)^{ \bigg(\frac{ - {tan}^(2)x}{1 + {tan}^(2)x} + \frac{ - {cot}^(2)x}{1 + {cot}^(2)x} \bigg)} \\ \end{gathered}`

We know,

`\begin{gathered} \purple{\rm :\longmapsto\:\boxed{\tt{ 1 + {tan}^(2)x = {sec}^(2)x}}} \\ \end{gathered}`

and

`\begin{gathered} \purple{\rm :\longmapsto\:\boxed{\tt{ 1 + {cot}^(2)x = {cosec}^(2)x}}} \\ \end{gathered}`

So, using this, we get

`\begin{gathered}\rm \: = \: \bigg( (25)/(777) \bigg)^{ \bigg(\frac{ - {tan}^(2)x}{{sec}^(2)x} + \frac{ - {cot}^(2)x}{{cosec}^(2)x} \bigg)} \\ \end{gathered}`

`\begin{gathered}\rm \: = \: \bigg( (25)/(777) \bigg)^{ \bigg(\frac{ - {sin}^(2)x}{{cos}^(2)x} * {cos}^(2)x + \frac{ - {cos}^(2)x}{{sin}^(2)x} * {sin}^(2)x \bigg)} \\ \end{gathered}`

`\begin{gathered}\rm \: = \: \bigg( (25)/(777) \bigg)^{\bigg( - {sin}^(2)x - {cos}^(2)x \bigg)} \\ \end{gathered}`

`\begin{gathered}\rm \: = \: \bigg( (25)/(777) \bigg)^{\bigg( -({sin}^(2)x + {cos}^(2)x) \bigg)} \\ \end{gathered}`

`\begin{gathered}\rm \: = \: \bigg( (25)/(777) \bigg)^(\bigg( -1 \bigg)) \\ \end{gathered}`

`\rm \: = \: (777)/(25)`

Hence,

`\begin{gathered} \\ \purple{\rm :\longmapsto\:\boxed{\tt{ \prod \limits^( \infty )_(n = 1) \bigg( (25)/(777) \bigg)^{ {( - 1)}^(n) \bigg( ( tan^(4n) (x) + 1)/( tan ^(2n) (x) ) \bigg)} = (777)/(25) \: }}} \\ \end{gathered}`