sin-1-1-5-cot-1-3-

Question Number 64773 by ankan0 last updated on 21/Jul/19

\sin^{- 1} (1 / \sqrt{5)} + \cot^{- 1} 3

Answered by Kunal12588 last updated on 21/Jul/19

\cot^{- 1} (a) = \sin^{- 1} (\frac{1}{\sqrt{1 + a^{2}}})

\cot^{- 1} 3 = \sin^{- 1} (\frac{1}{\sqrt{1 + 9}}) = \sin^{- 1} \frac{1}{\sqrt{10}}

\sin^{- 1} a + \sin^{- 1} b = \sin^{- 1} (a \sqrt{1 - b^{2}} + b \sqrt{1 - a^{2}})

\sin^{- 1} (1 / \sqrt{5}) + \cot^{- 1} 3

= \sin^{- 1} \frac{1}{\sqrt{5}} + \sin^{- 1} \frac{1}{\sqrt{10}}

= \sin^{- 1} (\frac{1}{\sqrt{5}} \sqrt{1 - \frac{1}{10}} + \frac{1}{\sqrt{10}} \sqrt{1 - \frac{1}{5}})

= \sin^{- 1} (\frac{3}{\sqrt{5} \sqrt{10}} + \frac{2}{\sqrt{10} \sqrt{5}})

= \sin^{- 1} (\frac{5}{5 \sqrt{2}})

= \sin^{- 1} (\frac{1}{\sqrt{2}})

= \frac{π}{4} = 45 °

Commented by John M. Kaloki last updated on 13/Aug/19

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