arctan(x+1/3)+arctan(x-1/3)=arctan(2)

Solución

arctan (x + \frac{1}{3}) + arctan (x - \frac{1}{3}) = arctan (2)

x = \frac{2}{3}

Pasos de solución

arctan (x + \frac{1}{3}) + arctan (x - \frac{1}{3}) = arctan (2)

Re-escribir usando identidades trigonométricas

arctan (\frac{x + \frac{1}{3} + x - \frac{1}{3}}{1 - ( x + \frac{1}{3} ) ( x - \frac{1}{3} )}) = arctan (2)

Aplicar propiedades trigonométricas inversas

\frac{x + \frac{1}{3} + x - \frac{1}{3}}{1 - ( x + \frac{1}{3} ) ( x - \frac{1}{3} )} = 2

Resolver

\frac{x + \frac{1}{3} + x - \frac{1}{3}}{1 - ( x + \frac{1}{3} ) ( x - \frac{1}{3} )} = 2 : x = - \frac{5}{3}, x = \frac{2}{3}

x = - \frac{5}{3}, x = \frac{2}{3}

Verificar las soluciones sustituyendo en la ecuación original

x = \frac{2}{3}

sin (θ) = 0, 64

cos (3 x) = cos (2 x)

csc (x^{2}) = \frac{4}{3}

2 cos (x) cos^{3} (x) + cos^{2} (x) - sin^{2} (x) = cos (x)

tan (θ) = \frac{2 3}{- 2}