Answer
$$\sin\theta=\frac{\sqrt{2x+1}}{|x+1|}$$
Work Step by Step
$$\cos\theta=\frac{x}{x+1}$$
According to the Pythagorean Identity
$$\sin^2\theta+\cos^2\theta=1$$
we can rewrite as $$\sin^2\theta=1-\cos^2\theta$$
Then, $$\sin\theta=\sqrt{1-\cos^2\theta}$$
This is the way we would be able to find an expression of $x$ for $\sin\theta$
1) Find $\cos^2\theta$
$$\cos^2\theta=\frac{x^2}{(x+1)^2}$$
2) Find $\sin^2\theta$
$$\sin^2\theta=1-\cos^2\theta$$
$$\sin^2\theta=1-\frac{x^2}{(x+1)^2}$$
$$\sin^2\theta=\frac{(x+1)^2-x^2}{(x+1)^2}$$
$$\sin^2\theta=\frac{(x+1-x)(x+1+x)}{(x+1)^2}$$ (for $a^2-b^2=(a-b)(a+b)$)
$$\sin^2\theta=\frac{1\times(2x+1)}{(x+1)^2}$$
$$\sin^2\theta=\frac{2x+1}{(x+1)^2}$$
3) Find $\sin\theta$
$$\sin\theta=\sqrt{\sin^2\theta}$$
$$\sin\theta=\frac{\sqrt{2x+1}}{\sqrt{(x+1)^2}}$$
$$\sin\theta=\frac{\sqrt{2x+1}}{|x+1|}$$ (since $\sqrt{a^2}=|a|$)
Unfortunately, we cannot eliminate the absolute value $||$ sign, since there is not enough information to know whether $(x+1)\gt0$ or $\lt0$.
