Question

(1-cos x)(1+secx)cot x=sen x

Answer 1

Respuesta:  $\mathrm{Verificar\:}\left(1-\cos \left(x\right)\right)\left(1+\sec \left(x\right)\right)\cot \left(x\right)=\sin \left(x\right):\quad \mathrm{Verdadero}$

Explicación paso a paso:

$\left(1-\cos \left(x\right)\right)\left(1+\sec \left(x\right)\right)\cot \left(x\right)=\sin \left(x\right)$

$\mathrm{Manipular\:el\:lado\:derecho}$

$\left(1-\cos \left(x\right)\right)\left(1+\sec \left(x\right)\right)\cot \left(x\right)$

$\mathrm{Expresar\:con\:seno,\:coseno}$

$=\left(1+\frac{1}{\cos \left(x\right)}\right)\left(1-\cos \left(x\right)\right)\frac{\cos \left(x\right)}{\sin \left(x\right)}$

$\mathrm{Simplificar}\:\left(1+\frac{1}{\cos \left(x\right)}\right)\left(1-\cos \left(x\right)\right)\frac{\cos \left(x\right)}{\sin \left(x\right)}:\quad \frac{\left(\cos \left(x\right)+1\right)\left(1-\cos \left(x\right)\right)}{\sin \left(x\right)}$

$=\frac{\left(1+\cos \left(x\right)\right)\left(1-\cos \left(x\right)\right)}{\sin \left(x\right)}$

$\mathrm{Expandir}\:\left(1-\cos \left(x\right)\right)\left(1+\cos \left(x\right)\right)=1-\cos ^2\left(x\right)$

$\mathrm{Usar\:la\:siguiente\:identidad:}\:1-\cos ^2\left(x\right)=\sin ^2\left(x\right)$

$=\frac{\sin ^2\left(x\right)}{\sin \left(x\right)}$

$\mathrm{Simplificar}$

$=\sin \left(x\right)$

$\sin \left(x\right)=\sin \left(x\right)$

$\mathrm{Se\:demostro\:que\:ambos\:lados\:pueden\:tomar\:la\:misma\:forma}$

$\Rightarrow \mathrm{Verdadero}$