Tan -117 - tan -1 113 - tan-1 29

tan^ 117+tan^ 1113=tan^ 1(17+1131 191)=tan^ 1(2090)=tan^ 1(29) {tan^ 1A tan^ 1B=tan^ 1(A+B1+AB)} . ...

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Principal Solution of Trigonometric Equation

tan -117 + ...

Question

tan $- 1 \frac{1}{7} + t a n^{-} 1 \frac{1}{13} = t a n^{-} 1 \frac{2}{9}$

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Solution

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{tan}^{- 1} (\frac{1}{7}) + {tan}^{- 1} (\frac{1}{13}) = {tan}^{- 1} (\frac{2}{9})

t a n^{- 1} 1 + t a n^{- 1} 2 + t a n^{- 1} 3 =

{tan}^{- 1} \frac{1}{4} + {tan}^{- 1} \frac{2}{9} = \frac{1}{2} {cos}^{- 1} \frac{3}{5}

t a n^{- 1} 1 + t a n^{- 1} 2 + t a n^{- 1} 3 = π

{s i n}^{- 1} x, {c o s}^{- 1} x, {t a n}^{- 1} x

{t a n}^{- 1} x + {t a n}^{- 1} y = {t a n}^{- 1} \frac{x + y}{1 - x y}

x y < 1

π + {t a n}^{- 1} \frac{x + y}{1 - x y}

x y > 1

{c o t}^{- 1} 9 + {c o s e c}^{1} \frac{\sqrt{41}}{4} = \frac{π}{4}

{t a n}^{- 1} \frac{1}{7} + {t a n}^{- 1} \frac{1}{13} = {t a n}^{- 1} \frac{2}{9}

{t a n}^{- 1} 2 + {t a n}^{- 1} 3 = 3 π / 4

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