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Simplify [tex]\frac{sinx + tanx}{cscx + cotx}[/tex]

I start with [tex]\frac{sinx + tanx}{cscx + cotx} = \frac{sinx + \frac{sinx}{cosx}}{\frac{1}{sinx} + \frac{cosx}{sinx}} [/tex]

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

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

I start with [tex]\frac{sinx + tanx}{cscx + cotx} = \frac{sinx + \frac{sinx}{cosx}}{\frac{1}{sinx} + \frac{cosx}{sinx}} [/tex]

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

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

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