(sin(x))/(cos(x))=(sin(x))/(1+cot^2(x))

Solución

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

Sin soluci \overset{o}{ˊ} n para x \in R

Pasos de solución

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

Restar

\frac{sin ( x )}{1 + cot ^{2} ( x )}

de ambos lados

\frac{sin ( x )}{cos ( x )} - \frac{sin ( x )}{1 + cot ^{2} ( x )} = 0

Simplificar

\frac{sin ( x )}{cos ( x )} - \frac{sin ( x )}{1 + cot ^{2} ( x )} : \frac{sin ( x ) ( cot ^{2} ( x ) + 1 ) - sin ( x ) cos ( x )}{cos ( x ) ( cot ^{2} ( x ) + 1 )}

\frac{sin ( x ) ( cot ^{2} ( x ) + 1 ) - sin ( x ) cos ( x )}{cos ( x ) ( cot ^{2} ( x ) + 1 )} = 0

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

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

Re-escribir usando identidades trigonométricas

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

Factorizar

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

sin (x) (1 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )} - cos (x)) = 0

Resolver cada parte por separado

sin (x) = 0 or 1 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )} - cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

1 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )} - cos (x) = 0 :

Sin solución

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

Siendo que la ecuación esta indefinida para:

2 πn, π + 2 πn

Sin soluci \overset{o}{ˊ} n para x \in R

3 + 8 cos (2 θ) = - 1

- 4 cos^{2} (x) + 6 cos (x) = 2

sec (θ) = - \frac{4}{3}

tan (2 x) = tan (201 8^{\circ})

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