verificar cos(x)(1+tan(x))^2=sec(x)+2sin(x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

cos (x) (1 + tan (x))^{2}

Expresar con seno, coseno

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

Expandir

(cos (x) + sin (x))^{2} : cos^{2} (x) + 2 cos (x) sin (x) + sin^{2} (x)

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

Re-escribir usando identidades trigonométricas

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

Re-escribir usando identidades trigonométricas

sec (x) + 2 sin (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

tan (θ) = \frac{sec ( θ )}{csc ( θ )}

\frac{sec ( θ )}{csc ^{2} ( θ )} = sec (θ) - cos (θ)

cos (3 β) = cos (β) (cos^{2} (β) - 3 sin^{2} (β))

sec^{2} (θ) cot^{2} (θ) - 1 = cot^{2} (θ)