Can someone show me how to simplify $\sin\left(\arccos\left(\frac{x}{x+1}\right)\right)$ I am needing help learning how to simplify the following equation: 
$$\sin\left(\arccos\left(\frac{x}{x+1}\right)\right)$$
Also any steps on how to get the answer would be greatly appreciated!
 A: as Rahul said... let $\arccos(\frac{x}{x+1})=\theta$ so $\cos\theta=\frac{x}{x+1}$ but you want $\sin\theta$ which is $\sqrt{1-\cos^2\theta}$.


*

*Remember $\arccos t$ is between $0$ and $\pi$, so $\sin(\arccos t)$ is positive. 


so the answer would be $$\sqrt{1-(\frac{x}{x+1})^2}=\sqrt{1-\frac{x^2}{(x+1)^2}}=\sqrt{\frac{(x+1)^2-x^2}{(x+1)^2}}=\sqrt{\frac{2x+1}{(x+1)^2}}=\frac{\sqrt{2x+1}}{x+1}.$$
A: Use $\sin^2\theta+\cos^2\theta = 1$
$$\begin{align} &~\sin\arccos\dfrac{x}{x+1}\\[1ex] =&~ \pm\sqrt{~1-\cos^2\arccos\dfrac x{x+1}~} \\[1ex] =&~ \pm\sqrt{~1-\dfrac{x^2}{(x+1)^2}~} \\[1ex] =&~ \pm\dfrac{\sqrt{~2x+1~}}{x+1}\end{align}$$
A: HINT
Put $\cos^{-1} \frac{x}{x+1} =t$
A: Draw a picture.  Let $ABC$ be a right triangle with right angle $\angle ABC$, and $\theta = \angle CAB$.  Assume $AB = x, AC = x+1$.  You can see that $$\cos \theta = \frac{AB}{AC} = \frac{x}{x+1}$$ By the Pythagorean theorem, $$BC = \sqrt{(x+1)^2 - x^2} = \sqrt{2x+1}$$  Now what's $\sin \theta$?
A: There is a subtle point: you first need to ensure that $x/(x+1)$ belongs to the domain of $\arccos$, that is,
$$
-1\le\frac{x}{x+1}\le 1
$$
With standard manipulations you get that your expression is defined for $x\ge-1/2$ (so, in particular, $x+1>0$).
Then, set $\alpha=\arccos(x/(x+1))$, so your expression is $\sin\alpha$, with $0\le\alpha\le\pi$. Hence,
$$
\sin\left(\arccos\frac{x}{x+1}\right)=
\sin\alpha=\sqrt{1-\cos^2\alpha}=\sqrt{1-\frac{x^2}{(x+1)^2}}
=\sqrt{\frac{2x+1}{(x+1)^2}}=\frac{\sqrt{2x+1}}{x+1}
$$
(the last step is justified by the fact that $x+1>0$).
