Value of the given expression ...

Question

If $$y=\tan^{-1}\left(\sqrt{\dfrac{1+\cos x}{1-\cos x}}\right)$$ then value of $(2x+14y)^3-343$ is ?
I reduced the equation as $y=\tan^{-1}\left(\dfrac{1+\cos x}{\sin x}\right)$ but I couldn't simplify it further. Thanks!

Answer 1

$$\cos x=\dfrac{1-\tan^2\dfrac x2}{1+\tan^2\dfrac x2}\iff\tan^2\dfrac x2=\dfrac{1-\cos x}{1+\cos x}$$

WLOG $0\le x\le\pi$

$$\implies\sqrt{\dfrac{1+\cos x}{1-\cos x}}=\cot\dfrac x2=\tan\left(\dfrac\pi2-\dfrac x2\right)$$

$$\implies\arctan\sqrt{\dfrac{1+\cos x}{1-\cos x}}=\dfrac\pi2-\dfrac x2$$

Answer 2

You can use a half-angle identity; specifically, $\sqrt{\frac{1 + \cos(x)}{1 - \cos(x)}} = \cot\left(\frac{x}{2}\right).$ So we have that $y = \tan^{-1}\left(\cot\left(\frac{x}{2}\right)\right) = \tan^{-1}\left(\tan\left(\frac{\pi}{2} - \frac{x}{2}\right)\right)$. This is simply equal to $\frac{\pi}{2} - \frac{x}{2},$ so $2y = \pi - x$ and $x = \pi - 2y.$

Performing the substitution (no numerical answer can possibly be solved for), we have the expression is equal to $\boxed{(2\pi + 10y)^{3} - 343}.$

Answer 3

Recall the bisection formulas: $$ \left|\sin\frac{x}{2}\right|=\sqrt{\frac{1-\cos x}{2}}, \qquad \left|\cos\frac{x}{2}\right|=\sqrt{\frac{1+\cos x}{2}} $$ so you have $$ \sqrt{\frac{1+\cos x}{1-\cos x}}=\left|\cot\frac{x}{2}\right| $$ For $x\in(0,\pi)$ we have $\cot(x/2)>0$, so $$ y=\arctan\cot\frac{x}{2} $$ and therefore $\tan y=\cot(x/2)=\tan(\pi/2-x/2)$ and therefore $y=\pi/2-x/2$.

For $x\in(\pi,2\pi)$ we have $\cot(x/2)<0$, so $$ y=-\arctan\cot\frac{x}{2} $$ and $\tan y=-\cot\frac{x}{2}=-\tan(\pi/2-x/2)=\tan(x/2-\pi/2)$, so $$ y=x/2-\pi/2 $$

In the case $x\in(0,\pi)$, we have $2x+14y=7\pi-5x$; in the case $x\in(\pi,2\pi)$ we have $2x+14y=9x-7\pi$.

The case $x=\pi$ is very easy: $y=0$.

It's more complicated if $x$ is supposed to be an arbitrary real number (excluding numbers of the form $2k\pi$, for $k$ an integer).

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If the answer has to be $9865$, you need either $$ 7\pi-5x=\sqrt[3]{10208},\qquad 0<x<\pi $$ or $$ 9x-7\pi=\sqrt[3]{10208},\qquad \pi<x<2\pi $$ The first case gives an acceptable solution $x\approx 0.05969$. The second case gives an acceptable solution $x\approx 4.85376$.