Formulas for Sum and Difference of Inverse Trigonometric Functions

  1. \[Si{n^{ – 1}}a + Si{n^{ – 1}}b = Si{n^{ – 1}}(a\sqrt {1 – {b^2}} + b\sqrt {1 – {a^2}} )\]
  2. \[Si{n^{ – 1}}a – Si{n^{ – 1}}b = Si{n^{ – 1}}(a\sqrt {1 – {b^2}} – b\sqrt {1 – {a^2}} )\]
  3. \[Co{s^{ – 1}}a + Co{s^{ – 1}}b = Co{s^{ – 1}}(ab – \sqrt {(1 – {a^2})(1 – {b^2})} )\]
  4. \[Co{s^{ – 1}}a – Co{s^{ – 1}}b = Co{s^{ – 1}}(ab + \sqrt {(1 – {a^2})(1 – {b^2})} )\]
  5. \[Ta{n^{ – 1}}a + Ta{n^{ – 1}}b = Ta{n^{ – 1}}\left( {\frac{{a + b}}{{1 – ab}}} \right)\]
  6. \[Ta{n^{ – 1}}a – Ta{n^{ – 1}}b = Ta{n^{ – 1}}\left( {\frac{{a – b}}{{1 + ab}}} \right)\]
  7. \[Si{n^{ – 1}}\left( {\frac{{2a}}{{1 + {a^2}}}} \right) = 2Ta{n^{ – 1}}a\]
  8. \[Co{s^{ – 1}}\left( {\frac{{1 – {a^2}}}{{1 + {a^2}}}} \right) = 2Ta{n^{ – 1}}a\]
  9. \[Ta{n^{ – 1}}\left( {\frac{{2a}}{{1 – {a^2}}}} \right) = 2Ta{n^{ – 1}}a\]
  10. \[Co{s^{ – 1}}a = Ta{n^{ – 1}}\left( {\frac{{\sqrt {1 – {a^2}} }}{a}} \right){\text{ }}0 < a \leqslant 1\]