sec^2(θ)+csc^2(θ)=4

Solución

sec^{2} (θ) + csc^{2} (θ) = 4

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

Pasos de solución

sec^{2} (θ) + csc^{2} (θ) = 4

Restar

4

de ambos lados

sec^{2} (θ) + csc^{2} (θ) - 4 = 0

Expresar con seno, coseno

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

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

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

Re-escribir usando identidades trigonométricas

1 - 4 cos^{2} (θ) sin^{2} (θ) = 0

Factorizar

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

(1 + 2 sin (θ) cos (θ)) (1 - 2 sin (θ) cos (θ)) = 0

Resolver cada parte por separado

1 + 2 sin (θ) cos (θ) = 0 or 1 - 2 sin (θ) cos (θ) = 0

1 + 2 sin (θ) cos (θ) = 0 : θ = \frac{3 π}{4} + πn

1 - 2 sin (θ) cos (θ) = 0 : θ = \frac{π}{4} + πn

Combinar toda las soluciones

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

cos (\frac{θ}{3}) = sin (θ)

sin (x) - cos (x) = \frac{1}{5}

tan (θ) = \frac{- 2}{3}

cos (θ) tan^{2} (θ) = 3 cos (θ)

cos (θ) = 0.56