Question

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

Trigonometric Functions
Write the trigonometric expression as an algebraic expression. $$\displaystyle{\cos{{\left({\arcsin{{x}}}−{\arctan{{2}}}{x}\right)}}}$$

2021-09-17
1. Let $$\displaystyle{u}={\arctan{{2}}}{x}$$ , then $$\displaystyle{\tan{{u}}}=\frac{{{2}{x}}}{{1}}=\frac{{{a}{d}{j}}}{{{o}{p}{p}}}.$$
2. $$\displaystyle{h}{y}{p}=\sqrt{{{1}+{4}{x}^{{2}}}}$$
3. Then $$\displaystyle{\cos{{u}}}=\frac{{{1}}}{\sqrt{{{1}-{4}{x}^{{2}}}}}={\cos{{\left({\arctan{{2}}}{x}\right)}}}$$
4. And $$\displaystyle{\sin{{u}}}=\frac{{{2}{x}}}{\sqrt{{{1}-{4}{x}^{{2}}}}}={\sin{{\left({\arctan{{2}}}{x}\right)}}}$$
5. Let $$\displaystyle{v}={\arccos{{x}}}$$ , then $$\displaystyle{\cos{{v}}}={x}.$$
6. $$\displaystyle{o}{p}{p}=\sqrt{{{1}-{x}^{{2}}}}$$
7. Then $$\displaystyle{\sin{{v}}}=\frac{\sqrt{{{1}-{x}^{{2}}}}}{{1}}={\sin{{\left({\arccos{{x}}}\right)}}}$$
8. $$\displaystyle{\cos{{\left({\arcsin{{x}}}-{\arctan{{2}}}{x}\right)}}}-{\cos{{\left({\arcsin{{x}}}\right)}}}{\cos{{\left({\arctan{{2}}}{x}\right)}}}+{\sin{{\left({\arcsin{{x}}}\right)}}}{\sin{{\left({\arctan{{2}}}{x}\right)}}}=$$
$$\displaystyle={\cos{{\left({\arcsin{{x}}}\right)}}}{\cos{{\left({\arctan{{2}}}{x}\right)}}}+{x}{\sin{{\left({\arctan{{2}}}{x}\right)}}}=\frac{{{2}{x}^{{2}}+\sqrt{{{1}-{x}^{{2}}}}}}{\sqrt{{{1}-{4}{x}^{{2}}}}}$$