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The equation is this:
[tex] \frac{sinx+ tanx }{1+secx } =sinx[/tex]
Recall these identities:
[tex]tanx= \frac{sinx}{cosx} [/tex]
[tex] secx= \frac{1}{cosx} [/tex]
Doing the necessary substitution, we get:
[tex] \frac{sinx+ \frac{sinx}{cosx} }{1+ \frac{1}{cosx} } =sinx[/tex]
Combining, we will have:
[tex]( \frac{sinxcosx+sinx}{cosx}) /( \frac{cosx+1}{cosx})=sinx [/tex]
When we divide, we multiply the numerator by the reciprocal of the denominator, we get:
[tex] \frac{sinxcosx+sinx}{cosx}* \frac{cosx}{cosx+1}=sinx [/tex]
We cancel cosx, then we get:
[tex] \frac{sinxcosx+sinx}{cosx+1}=sinx[/tex]
Factor sinx from the numerator,
[tex] \frac{sinx(cosx+1)}{cosx+1}=sinx[/tex]
Cancel cosx+1,
[tex]sinx=sinx[/tex]
Yay.
Hope that helps.
