verificar (1+csc(θ))/(sec(θ))=cos(θ)+cot(θ)

Solución

verificar

\frac{1 + csc ( θ )}{sec ( θ )} = cos (θ) + cot (θ)

Verdadero

Pasos de solución

\frac{1 + csc ( θ )}{sec ( θ )} = cos (θ) + cot (θ)

Manipular el lado izquierdo

cos (θ) + cot (θ)

Expresar con seno, coseno

= \frac{cos ( θ ) + cos ( θ ) sin ( θ )}{sin ( θ )}

Factorizar

\frac{cos ( θ ) + cos ( θ ) sin ( θ )}{sin ( θ )} : \frac{cos ( θ ) ( 1 + sin ( θ ) )}{sin ( θ )}

= \frac{( 1 + sin ( θ ) ) cos ( θ )}{sin ( θ )}

Re-escribir usando identidades trigonométricas

\frac{csc ( θ ) + 1}{sec ( θ )}

= \frac{1 + csc ( θ )}{sec ( θ )}

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

\frac{cos ( x )}{sec ( 2 x )} - \frac{sin ( x )}{csc ( 2 x )} = cos (3 x)

cot (a) + \frac{sin ( a )}{1 + cos ( a )} = csc (a)

sec^{2} (x) - cot^{2} (\frac{π}{2} - x) = 1

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

sin (2 x) = - 2 sin (- x) cos (- x)