Question

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

Trigonometric Functions
ANSWERED
asked 2021-09-16
Write the trigonometric expression as an algebraic expression. \(\displaystyle{\cos{{\left({\arcsin{{x}}}−{\arctan{{2}}}{x}\right)}}}\)

Expert Answers (1)

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. \(\displaystyle{\cos{{\left({\arcsin{{x}}}-{\arctan{{2}}}{x}\right)}}}-{\cos{{\left({\arcsin{{x}}}\right)}}}{\cos{{\left({\arctan{{2}}}{x}\right)}}}+{\sin{{\left({\arcsin{{x}}}\right)}}}{\sin{{\left({\arctan{{2}}}{x}\right)}}}=\)
\(\displaystyle={\cos{{\left({\arcsin{{x}}}\right)}}}{\cos{{\left({\arctan{{2}}}{x}\right)}}}+{x}{\sin{{\left({\arctan{{2}}}{x}\right)}}}=\frac{{{2}{x}^{{2}}+\sqrt{{{1}-{x}^{{2}}}}}}{\sqrt{{{1}-{4}{x}^{{2}}}}}\)
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