2.3 Inverse Trigonometry Function Identities 反三角函数的恒等式 - 独中课程 | 线上补习

高三数学 | 高级数学

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2.3 Inverse Trigonometry Function Identities 反三角函数的恒等式

Prove the following inverse trigonometry statements
试证明下列各反三角函数式

1) $\tan^{-1}{\frac{4}{3}}+\sin^{-1}{\frac{12}{13}}=\cos^{-1}{\left(-\frac{33}{65}\right)}$

2) $\sin^{-1}{\frac{4}{5}}+\cos^{-1}{\frac{12}{13}}=\tan^{-1}{\frac{63}{16}}$

3) ${cos}^{-1}{\frac{3}{5}}+{tan}^{-1}{\frac{5}{12}}={tan}^{-1}{\frac{63}{16}}$

4) ${cos}^{-1}{\frac{3}{5}}+{sin}^{-1}{\frac{12}{13}}=\pi-{sin}^{-1}{\frac{56}{65}}$

5) $2{tan}^{-1}{\frac{1}{3}}={tan}^{-1}{\frac{3}{4}}$

6) $\sin^{-1}{\frac{3}{5}}+\sin^{-1}{\frac{8}{17}}=\sin^{-1}{\frac{77}{85}}$