teddytinotendak

2022-07-18

simplify (sinx/(1+sinx)) + ((1+sinx)/cosx)

Eliza Beth13

To simplify the expression $\frac{\mathrm{sin}\left(x\right)}{1+\mathrm{sin}\left(x\right)}+\frac{1+\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$, we'll start by finding a common denominator for the two fractions.
$\frac{\mathrm{sin}\left(x\right)}{1+\mathrm{sin}\left(x\right)}+\frac{1+\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$
Let's find the common denominator for the two fractions, which is $\mathrm{cos}\left(x\right)\left(1+\mathrm{sin}\left(x\right)\right)$.
$\frac{\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)}{\mathrm{cos}\left(x\right)\left(1+\mathrm{sin}\left(x\right)\right)}+\frac{\left(1+\mathrm{sin}\left(x\right)\right)·\left(1+\mathrm{sin}\left(x\right)\right)}{\mathrm{cos}\left(x\right)\left(1+\mathrm{sin}\left(x\right)\right)}$
Now, we can combine the fractions over the common denominator.
$\frac{\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)+\left(1+\mathrm{sin}\left(x\right){\right)}^{2}}{\mathrm{cos}\left(x\right)\left(1+\mathrm{sin}\left(x\right)\right)}$
Expanding $\left(1+\mathrm{sin}\left(x\right){\right)}^{2}$ gives us:
$\frac{\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)+1+2\mathrm{sin}\left(x\right)+{\mathrm{sin}}^{2}\left(x\right)}{\mathrm{cos}\left(x\right)\left(1+\mathrm{sin}\left(x\right)\right)}$
Simplifying the expression further, we have:
$\frac{\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)+1+2\mathrm{sin}\left(x\right)+{\mathrm{sin}}^{2}\left(x\right)}{\mathrm{cos}\left(x\right)+\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)}$
Now, let's simplify the numerator. Combining like terms, we get:
$\frac{{\mathrm{sin}}^{2}\left(x\right)+2\mathrm{sin}\left(x\right)+\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)+1}{\mathrm{cos}\left(x\right)+\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)}$
Grouping the terms with $\mathrm{sin}\left(x\right)$ together:
$\frac{{\mathrm{sin}}^{2}\left(x\right)+3\mathrm{sin}\left(x\right)+1}{\mathrm{cos}\left(x\right)+\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)}$
Factoring the numerator:
$\frac{\left(\mathrm{sin}\left(x\right)+1\right)\left(\mathrm{sin}\left(x\right)+1\right)}{\mathrm{cos}\left(x\right)+\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)}$
Simplifying the expression even further:
$\frac{\left(\mathrm{sin}\left(x\right)+1{\right)}^{2}}{\mathrm{cos}\left(x\right)+\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)}$
Therefore, the simplified form of the expression $\frac{\mathrm{sin}\left(x\right)}{1+\mathrm{sin}\left(x\right)}+\frac{1+\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$ is $\frac{\left(\mathrm{sin}\left(x\right)+1{\right)}^{2}}{\mathrm{cos}\left(x\right)+\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)}$.

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