Simplify quotient of trignometric functions

[username12345]

Simplify \frac{sinx + tanx}{cscx + cotx}

I start with \frac{sinx + tanx}{cscx + cotx} = \frac{sinx + \frac{sinx}{cosx}}{\frac{1}{sinx} + \frac{cosx}{sinx}}

At this point I am stuck, I cannot see how we then get from \frac{sinx + \frac{sinx}{cosx}}{\frac{1}{cosx} + \frac{cosx}{sinx}} = \frac{sinx cosx + sin x}{1} \times \frac{sin x}{cos x}

What happened to the \frac{1}{cos x} from the denominator? I thought that would become \frac{cos x}{1} to give (sinx + \frac{sinx}{cosx}) \times ({\frac{cosx}{1} + \frac{sin x}{cos x})

 

[username12345]

I think I figured it out. Needed to simplify the numerator and denominator into a single fraction, then invert the denominator and multiply. This was the step I was missing. Also needed to factorise the result of the multiplication.