cos(x)(tan(x)+(cos(x))/(sin(x)))=sqrt(2)

Solución

cos (x) (tan (x) + \frac{cos ( x )}{sin ( x )}) = 2

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Pasos de solución

cos (x) (tan (x) + \frac{cos ( x )}{sin ( x )}) = 2

Restar

2

de ambos lados

cos (x) (tan (x) + \frac{cos ( x )}{sin ( x )}) - 2 = 0

Simplificar

cos (x) (tan (x) + \frac{cos ( x )}{sin ( x )}) - 2 : \frac{cos ( x ) ( tan ( x ) sin ( x ) + cos ( x ) ) - 2 sin ( x )}{sin ( x )}

\frac{cos ( x ) ( tan ( x ) sin ( x ) + cos ( x ) ) - 2 sin ( x )}{sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos (x) (tan (x) sin (x) + cos (x)) - 2 sin (x) = 0

Re-escribir usando identidades trigonométricas

- 2 sin (x) + 1 = 0

Mover

1

al lado derecho

- 2 sin (x) = - 1

Dividir ambos lados entre

- 2

sin (x) = \frac{2}{2}

Soluciones generales para

sin (x) = \frac{2}{2}

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

2 cos (2 x) + sin (x) = \frac{1}{2}

2 cos^{2} (x) + cos (x) + 1 = 0

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

- 2 + 3 csc (x) = 5

2 sin (2 x) = sin (4 x), 0 \leq x \leq π