[tex]\large\underline{\sf{Solution-}} [/tex]
Given expression is
[tex]\rm :\longmapsto\:\prod \limits^{ \infty }_{n = 1} \bigg( \dfrac{25}{777} \bigg)^{ {( - 1)}^{n} \bigg( \dfrac{ tan^{4n} (x) + 1}{ tan ^{2n} (x) } \bigg)}[/tex]
can be rewritten as
[tex]\rm \: = \: \prod \limits^{ \infty }_{n = 1} \bigg( \dfrac{25}{777} \bigg)^{ {( - 1)}^{n} \bigg( \dfrac{ tan^{4n} (x)}{ tan ^{2n} (x)} + \dfrac{1}{ {tan}^{2n} (x)} \bigg)}[/tex]
[tex]\rm \: = \: \prod \limits^{ \infty }_{n = 1} \bigg( \dfrac{25}{777} \bigg)^{ {( - 1)}^{n} \bigg( {tan}^{2n}(x) + {cot}^{2n}(x) \bigg)}[/tex]
can be further rewritten as
[tex]\begin{gathered}\rm \: = \: \prod \limits^{ \infty }_{n = 1} \bigg( \dfrac{25}{777} \bigg)^{ {( - 1)}^{n} \bigg( {( {tan}^{2}x) }^{n} + {( {cot}^{2} x)}^{n} \bigg)} \\ \end{gathered} [/tex]
[tex]\begin{gathered}\rm \: = \: \prod \limits^{ \infty }_{n = 1} \bigg( \dfrac{25}{777} \bigg)^{\bigg( { {( - 1)}^{n} ( {tan}^{2}x) }^{n} + {( - 1)}^{n} {( {cot}^{2} x)}^{n} \bigg)} \\ \end{gathered}[/tex]
[tex]\begin{gathered}\rm \: = \: \prod \limits^{ \infty }_{n = 1} \bigg( \dfrac{25}{777} \bigg)^{ \bigg( {( { - tan}^{2}x) }^{n} + {( { - cot}^{2} x)}^{n} \bigg)} \\ \end{gathered} [/tex]
[tex]\begin{gathered}\rm \: = \: \bigg( \dfrac{25}{777} \bigg)^{ \bigg( {( { - tan}^{2}x) }^{1} + {( { - cot}^{2} x)}^{1} \bigg)} \times \bigg( \dfrac{25}{777} \bigg)^{ \bigg( {( { - tan}^{2}x) }^{2} + {( { - cot}^{2} x)}^{2} \bigg)} \times \bigg( \dfrac{25}{777} \bigg)^{ \bigg( {( { - tan}^{2}x) }^{3} + {( { - cot}^{2} x)}^{3} \bigg)} - - - \\ \end{gathered} [/tex]
[tex]\rm \: = \: \bigg( \dfrac{25}{777} \bigg)^{ \bigg( - {tan}^{2}x - {cot}^{2}x + {tan}^{4}x + {cot}^{4}x - {tan}^{6}x - {cot}^{6}x + - - \bigg)}[/tex]
[tex]\begin{gathered}\rm \: = \: \bigg( \dfrac{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} [/tex]
Using Sum of infinite GP series,
[tex]\begin{gathered} \purple{\rm :\longmapsto\:\boxed{\tt{ S_ \infty \: = \: \frac{a}{1 - r} \: where \: |r| < 1}}} \\ \end{gathered}[/tex]
So, using this,
[tex]\begin{gathered}\rm \: = \: \bigg( \dfrac{25}{777} \bigg)^{ \bigg(\dfrac{ - {tan}^{2}x}{1 + {tan}^{2}x} + \dfrac{ - {cot}^{2}x}{1 + {cot}^{2}x} \bigg)} \\ \end{gathered} [/tex]
We know,
[tex]\begin{gathered} \purple{\rm :\longmapsto\:\boxed{\tt{ 1 + {tan}^{2}x = {sec}^{2}x}}} \\ \end{gathered} [/tex]
and
[tex]\begin{gathered} \purple{\rm :\longmapsto\:\boxed{\tt{ 1 + {cot}^{2}x = {cosec}^{2}x}}} \\ \end{gathered} [/tex]
So, using this, we get
[tex]\begin{gathered}\rm \: = \: \bigg( \dfrac{25}{777} \bigg)^{ \bigg(\dfrac{ - {tan}^{2}x}{{sec}^{2}x} + \dfrac{ - {cot}^{2}x}{{cosec}^{2}x} \bigg)} \\ \end{gathered} [/tex]
[tex]\begin{gathered}\rm \: = \: \bigg( \dfrac{25}{777} \bigg)^{ \bigg(\dfrac{ - {sin}^{2}x}{{cos}^{2}x} \times {cos}^{2}x + \dfrac{ - {cos}^{2}x}{{sin}^{2}x} \times {sin}^{2}x \bigg)} \\ \end{gathered} [/tex]
[tex]\begin{gathered}\rm \: = \: \bigg( \dfrac{25}{777} \bigg)^{\bigg( - {sin}^{2}x - {cos}^{2}x \bigg)} \\ \end{gathered} [/tex]
[tex]\begin{gathered}\rm \: = \: \bigg( \dfrac{25}{777} \bigg)^{\bigg( -({sin}^{2}x + {cos}^{2}x) \bigg)} \\ \end{gathered} [/tex]
[tex]\begin{gathered}\rm \: = \: \bigg( \dfrac{25}{777} \bigg)^{\bigg( -1 \bigg)} \\ \end{gathered} [/tex]
[tex]\rm \: = \: \dfrac{777}{25}[/tex]
Hence,
[tex]\begin{gathered} \\ \purple{\rm :\longmapsto\:\boxed{\tt{ \prod \limits^{ \infty }_{n = 1} \bigg( \frac{25}{777} \bigg)^{ {( - 1)}^{n} \bigg( \frac{ tan^{4n} (x) + 1}{ tan ^{2n} (x) } \bigg)} = \frac{777}{25} \: }}} \\ \end{gathered} [/tex]
