3.9 Property 9

2.1 ${\sin ^{ - 1}}x$:
2.2 ${\cos ^{ - 1}}x$:
2.3 ${\tan ^{ - 1}}x$:
2.4 ${\text{cose}}{{\text{c}}^{ - 1}}x$:
2.5 ${\sec^{ - 1}}x$
2.6 ${\cot^{ - 1}}x$
2.7 Summary

3.1 Property 1
3.2 Property 2
3.3 Property 3
3.4 Property 4
3.5 Property 5
3.6 Property 6
3.7 Property 7
3.8 Property 8
3.9 Property 9
3.10 Property 10
3.11 Property 11
3.12 Property 12
3.13 Property 13

$$\begin{equation} \begin{aligned} (i){\sin ^{ - 1}}x = {\cos ^{ - 1}}\left( {\sqrt {1 - {x^2}} } \right) = {\tan ^{ - 1}}\left( {\frac{x}{{\sqrt {1 - {x^2}} }}} \right) = {\cot ^{ - 1}}\left( {\frac{{\sqrt {1 - {x^2}} }}{x}} \right) = {\sec ^{ - 1}}\left( {\frac{1}{{\sqrt {1 - {x^2}} }}} \right) = \cos e{c^{ - 1}}\left( {\frac{1}{x}} \right) \\ (ii){\cos ^{ - 1}}x = {\sin ^{ - 1}}\left( {\sqrt {1 - {x^2}} } \right) = {\tan ^{ - 1}}\left( {\frac{{\sqrt {1 - {x^2}} }}{x}} \right) = {\cot ^{ - 1}}\left( {\frac{x}{{\sqrt {1 - {x^2}} }}} \right) = se{c^{ - 1}}\left( {\frac{1}{x}} \right) = \cos e{c^{ - 1}}\left( {\frac{1}{{\sqrt {1 - {x^2}} }}} \right) \\ (iii){\tan ^{ - 1}}x = {\sin ^{ - 1}}\left( {\frac{x}{{\sqrt {1 + {x^2}} }}} \right) = {\cos ^{ - 1}}\left( {\frac{1}{{\sqrt {1 + {x^2}} }}} \right) = {\cot ^{ - 1}}\left( {\frac{1}{x}} \right) = \cos e{c^{ - 1}}\left( {\frac{{\sqrt {1 + {x^2}} }}{x}} \right) = {\sec ^{ - 1}}\left( {\sqrt {1 + {x^2}} } \right) \\\end{aligned} \end{equation} $$