Question

$\large \displaystyle\sf \prod^{ \infty }_{n = 1} \bigg( \frac{25}{777} \bigg)^{ {( - 1)}^{n} \bigg( \frac{ \tan^{4n} (x) + 1}{ \tan ^{2n} (x) } \bigg) }$​

Answer 1

Step-by-step explanation:

We have,

$\tt{\displaystyle\,P=\prod^{\infty}_{n=1}\,\left(\dfrac{25}{777}\right)^{\displaystyle(-1)^n\left(\dfrac{tan^{4n}(x)+1}{tan^{2n}(x)}\right)}}$

$\tt{\displaystyle\,\implies\,P=\prod^{\infty}_{n=1}\,\left(\dfrac{25}{777}\right)^{\displaystyle(-1)^n\left(tan^{2n}(x)+cot^{2n}(x)\right)}}$

$\tt{\displaystyle\,\implies\,P=\prod^{\infty}_{n=1}\,\left(\dfrac{25}{777}\right)^{\displaystyle(-1)^n\left\{\left(tan^{2}(x)\right)^n+\left(cot^{2}(x)\right)^n\right\}}}$

$\tt{\displaystyle\,\implies\,P=\prod^{\infty}_{n=1}\,\left(\dfrac{25}{777}\right)^{\displaystyle\left\{(-1)^n\cdot\left(tan^{2}(x)\right)^n+(-1)^n\cdot\left(cot^{2}(x)\right)^n\right\}}}$

$\tt{\displaystyle\,\implies\,P=\prod^{\infty}_{n=1}\,\left(\dfrac{25}{777}\right)^{\displaystyle\left\{\left(-tan^{2}(x)\right)^n+\left(-cot^{2}(x)\right)^n\right\}}}$

Now, consider the proposition,

$\bigstar\,\,\sf{\displaystyle\prod^{\infty}_{m=1}\big(k\big)^{\displaystyle\,a^{n}}}$

$\mapsto\,\,\sf{\displaystyle\big(k\big)^{\displaystyle\,a}\cdot\big(k\big)^{\displaystyle\,a^2}\cdot\big(k\big)^{\displaystyle\,a^3}\cdot\big(k\big)^{\displaystyle\,a^4}\cdots}$

$\mapsto\,\,\sf{\displaystyle\big(k\big)^{\displaystyle\,a+a^2+a^3+a^4+\cdots}}$

$\mapsto\,\,\sf{\displaystyle\big(k\big)^{\displaystyle\,\sum^{\infty}_{m=1}\,a^m}}$

So, our series may be rewritten as,

$\tt{\displaystyle\,\implies\,P=\left(\dfrac{25}{777}\right)^{\displaystyle\sum^{\infty}_{n=1}\left\{\left(-tan^{2}(x)\right)^n+\left(-cot^{2}(x)\right)^n\right\}}}$

Let us now consider the sum,

$\tt{\displaystyle\,S=\sum^{\infty}_{n=1}\left\{\left(-tan^{2}(x)\right)^n+\left(-cot^{2}(x)\right)^n\right\}}$

$\tt{\displaystyle\,\implies\,S=\sum^{\infty}_{n=1}\left(-tan^{2}(x)\right)^n+\sum^{\infty}_{n=1}\left(-cot^{2}(x)\right)^n}$

$\bf{Put\,\,\,\, \alpha=-tan^2(x)\,\,\,\,\,\,and\,\,\,\,\,\,\beta=-cot^2(x)}$

So,

$\tt{\displaystyle\,\implies\,S=\sum^{\infty}_{n=1}\left(\alpha\right)^n+\sum^{\infty}_{n=1}\left(\beta\right)^n}$

$\tt{\displaystyle\,\implies\,S=\sum^{\infty}_{n=1}\left(\alpha\right)^n+\sum^{\infty}_{n=1}\left(\beta\right)^n}$

These are GPs

So,

$\tt{\displaystyle\,\implies\,S=\dfrac{\alpha}{1-\alpha}+\dfrac{\beta}{1-\beta}}$

Putting the values of $\alpha$ and $\beta$

$\tt{\displaystyle\,\implies\,S=\dfrac{-tan^2(x)}{1+tan^2(x)}+\dfrac{-cot^2(x)}{1+cot^2(x)}}$

$\tt{\displaystyle\,\implies\,S=-\left(\dfrac{tan^2(x)}{sec^2(x)}+\dfrac{cot^2(x)}{cosec^2(x)}\right)}$

$\tt{\displaystyle\,\implies\,S=-\left(sin^2(x)+cos^2(x)\right)}$

$\tt{\displaystyle\,\implies\,S=-\left(1\right)}$

$\tt{\displaystyle\,\implies\,S=-1}$

$\tt{\displaystyle\,\therefore\sum^{\infty}_{n=1}\left\{\left(-tan^{2}(x)\right)^n+\left(-cot^{2}(x)\right)^n\right\}=-1}$

So, our series becomes,

$\tt{\displaystyle\,\implies\,P=\left(\dfrac{25}{777}\right)^{\displaystyle\sum^{\infty}_{n=1}\left\{\left(-tan^{2}(x)\right)^n+\left(-cot^{2}(x)\right)^n\right\}}=\left(\dfrac{25}{777}\right)^{-1}}$

$\tt{\displaystyle\,\implies\,P=\dfrac{777}{25}}$

Hope this will help you!! :):)