Multiply and divide by the conjugate of

Multiplying the fraction by $\sqrt{1+\cos\left(x\right)}$

Apply the property of power of a product in reverse: $a^n\cdot b^n=(a\cdot b)^n$

Multiply the single term $1+\cos\left(x\right)$ by each term of the polynomial $\left(1-\cos\left(x\right)\right)$

Solve the product $-\cos\left(x\right)\left(1+\cos\left(x\right)\right)$

We can solve the integral $\int\frac{\sin\left(x\right)}{\sqrt{1-\cos\left(x\right)}}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $\sqrt{1-\cos\left(x\right)}$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

Isolate $dx$ in the previous equation

The integral of a constant is equal to the constant times the integral's variable

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$