# Write the trigonometric expression as an algebraic expression. cos(arcsinx-arctan2x)

Write the trigonometric expression as an algebraic expression. $\mathrm{cos}\left(\mathrm{arcsin}x-\mathrm{arctan}2x\right)$
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odgovoreh
1. Let $u=\mathrm{arctan}2x$ , then $\mathrm{tan}u=\frac{2x}{1}=\frac{adj}{opp}.$
2. $hyp=\sqrt{1+4{x}^{2}}$
3. Then $\mathrm{cos}u=\frac{1}{\sqrt{1-4{x}^{2}}}=\mathrm{cos}\left(\mathrm{arctan}2x\right)$
4. And $\mathrm{sin}u=\frac{2x}{\sqrt{1-4{x}^{2}}}=\mathrm{sin}\left(\mathrm{arctan}2x\right)$
5. Let $v=\mathrm{arccos}x$ , then $\mathrm{cos}v=x.$
6. $opp=\sqrt{1-{x}^{2}}$
7. Then $\mathrm{sin}v=\frac{\sqrt{1-{x}^{2}}}{1}=\mathrm{sin}\left(\mathrm{arccos}x\right)$
8. $\mathrm{cos}\left(\mathrm{arcsin}x-\mathrm{arctan}2x\right)-\mathrm{cos}\left(\mathrm{arcsin}x\right)\mathrm{cos}\left(\mathrm{arctan}2x\right)+\mathrm{sin}\left(\mathrm{arcsin}x\right)\mathrm{sin}\left(\mathrm{arctan}2x\right)=$
$=\mathrm{cos}\left(\mathrm{arcsin}x\right)\mathrm{cos}\left(\mathrm{arctan}2x\right)+x\mathrm{sin}\left(\mathrm{arctan}2x\right)=\frac{2{x}^{2}+\sqrt{1-{x}^{2}}}{\sqrt{1-4{x}^{2}}}$