MHB How to Find x Given sin^{-1}(x) + cos^{-1}(1/√x) = 0?

[Amer]

Find x such that

sin^{-1} (x) + cos^{-1}\left( \frac{1}{\sqrt{x}}\right) = 0

 

[Siron]

Find x such that

sin^{-1} (x) + cos^{-1}\left( \frac{1}{\sqrt{x}}\right) = 0

First let's take the $\sin$ of both sides:
$$\sin\left[\mbox{arc}\sin(x)+\mbox{arc}\cos\left(\frac{1}{ \sqrt{x}}\right)\right]=\sin(0)$$
$$\Leftrightarrow \sin\left[\mbox{arc}\sin(x)\right]\cos\left[\mbox{arc}\cos\left(\frac{1}{ \sqrt{x}}\right)\right]+\sin\left[\mbox{arc}\cos\left(\frac{1}{ \sqrt{x}}\right)\right]\cos\left[\mbox{arc}\sin(x)\right]=0$$
$$\Leftrightarrow x\left(\frac{1}{\sqrt{x}}\right)+\sqrt{1-\left(\frac{1}{\sqrt{x}}\right)^2}\sqrt{1-x^2}=0$$
$$\Leftrightarrow \sqrt{x}+\sqrt{1-\frac{1}{x}}\sqrt{1-x^2}=0$$
$$\Leftrightarrow \sqrt{x}+\sqrt{\frac{x-1}{x}}\sqrt{1-x^2}=0$$
$$\Leftrightarrow \ldots$$

Now you have to solve an irrational equation.

 

[Amer]

Thanks