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verificar (cot(θ)-csc(θ))/(cot(θ)+csc(θ))=(1-2cos(θ)+cos^2(θ))/(-sin^2(θ))

Solución

verificar

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )} = \frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{- sin ^{2} ( θ )}

Verdadero

Pasos de solución

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )} = \frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{- sin ^{2} ( θ )}

Manipular el lado derecho

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )}

Expresar con seno, coseno

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

Manipular el lado izquierdo

\frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{- sin ^{2} ( θ )}

Re-escribir usando identidades trigonométricas

= \frac{cos ( θ ) - 1}{cos ( θ ) + 1}

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

sin^{2} (\frac{θ}{2}) = \frac{csc ( θ ) - cot ( θ )}{2 csc ( θ )}

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

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

cos (x) + cos (x) cot^{2} (x) = cot (x) csc (x)