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verificar sin(x)cos(x)+sin^3(x)sec(x)=tan(x)

Solución

verificar

sin (x) cos (x) + sin^{3} (x) sec (x) = tan (x)

Verdadero

Pasos de solución

sin (x) cos (x) + sin^{3} (x) sec (x) = tan (x)

Manipular el lado derecho

sin (x) cos (x) + sin^{3} (x) sec (x)

Expresar con seno, coseno

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

Factorizar

\frac{sin ^{3} ( x ) + cos ^{2} ( x ) sin ( x )}{cos ( x )} : \frac{sin ( x ) ( sin ^{2} ( x ) + cos ^{2} ( x ) )}{cos ( x )}

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica básica:

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

= tan (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

- tan (θ) cos (θ) = sin (- θ)

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

sin^{2} (2 x) = \frac{1}{2} (1 - cos (4 x))

sec (t) - \frac{cos ( t )}{1 + sin ( t )} = tan (t)