cos^2(x)-sin^2(x)=sec^2(x)-csc^2(x)

Solución

cos^{2} (x) - sin^{2} (x) = sec^{2} (x) - csc^{2} (x)

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

Pasos de solución

cos^{2} (x) - sin^{2} (x) = sec^{2} (x) - csc^{2} (x)

Restar

sec^{2} (x) - csc^{2} (x)

de ambos lados

cos^{2} (x) - sin^{2} (x) - sec^{2} (x) + csc^{2} (x) = 0

Expresar con seno, coseno

\frac{cos ^{2} ( x ) - sin ^{2} ( x ) - cos ^{2} ( x ) sin ^{4} ( x ) + cos ^{4} ( x ) sin ^{2} ( x )}{cos ^{2} ( x ) sin ^{2} ( x )} = 0

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

cos^{2} (x) - sin^{2} (x) - cos^{2} (x) sin^{4} (x) + cos^{4} (x) sin^{2} (x) = 0

Factorizar

cos^{2} (x) - sin^{2} (x) - cos^{2} (x) sin^{4} (x) + cos^{4} (x) sin^{2} (x) : (1 + cos^{2} (x) sin^{2} (x)) (cos (x) + sin (x)) (cos (x) - sin (x))

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

Resolver cada parte por separado

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

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

Sin solución

cos (x) + sin (x) = 0 : x = \frac{3 π}{4} + πn

cos (x) - sin (x) = 0 : x = \frac{π}{4} + πn

Combinar toda las soluciones

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

2 - 3 sin (x) = 0

y = cos (x), x

\frac{cos ( θ )}{sin ( θ )} = 0

sin (2 x) = sin (x), 0 \leq x < 2 π

tan (θ) = \frac{5}{5 3}