Question

demostrar las siguientes identidades trigonométricas

ayúdeme no lo entiendo por favor ​

Answer 1

Respuesta:

1)

$\mathrm{\:}\frac{\sin \left(x\right)\left(1-\cot \left(x\right)\right)}{\sin \left(x\right)-\cos \left(x\right)}=1$

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

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

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

=1

2)

$\sec ^2\left(u\right)\left(\csc ^2\left(u\right)-1\right)=\csc ^2\left(u\right)$

$=\left(-1+\csc ^2\left(u\right)\right)\left(1+\tan ^2\left(u\right)\right)$

$=\left(-1\right)\cdot \:1+\left(-1\right)\tan ^2\left(u\right)+\csc ^2\left(u\right)\cdot \:1+\csc ^2\left(u\right)\tan ^2\left(u\right)$

$=-1\cdot \:1-1\cdot \tan ^2\left(u\right)+1\cdot \csc ^2\left(u\right)+\csc ^2\left(u\right)\tan ^2\left(u\right)$

$-1+\csc ^2\left(u\right)-\tan ^2\left(u\right)+\csc ^2\left(u\right)\tan ^2\left(u\right)$

$-\tan ^2\left(u\right)+\csc ^2\left(u\right)\tan ^2\left(u\right)-1=0$

$\csc ^2\left(u\right)$  verdadero

3)$\frac{1}{\cos \left(x\right)\csc \left(x\right)}=\tan \left(x\right)$

$Expresar\:con\:seno,\:coseno\\\frac{1}{\cos \left(x\right)\frac{1}{\sin \left(x\right)}}:\quad \frac{\sin \left(x\right)}{\cos \left(x\right)}$

$\frac{\sin \left(x\right)}{\cos \left(x\right)}$

$\tan \left(x\right) verdadero$

4)

$\frac{\cos ^2\left(x\right)}{\sin ^2\left(x\right)}+\tan \left(x\right)\cot \left(x\right)=\csc ^2\left(x\right)$

$\frac{\cos ^2\left(x\right)}{\sin ^2\left(x\right)}+\frac{\cos \left(x\right)}{\sin \left(x\right)}\cdot \frac{\sin \left(x\right)}{\cos \left(x\right)}$

-----

$\frac{\cos \left(x\right)}{\sin \left(x\right)}\cdot \frac{\sin \left(x\right)}{\cos \left(x\right)}=1$  esto se suma como 1

$\frac{\cos ^2\left(x\right)}{\sin ^2\left(x\right)}+1$

$\frac{\cos ^2\left(x\right)}{\sin ^2\left(x\right)}+\frac{1\cdot \sin ^2\left(x\right)}{\sin ^2\left(x\right)}$

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

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

$\frac{1}{\sin ^2\left(x\right)}$

$\frac{1}{\left(\frac{1}{\csc \left(x\right)}\right)^2}$

$\mathrm{Simplificar}\:\frac{1}{\left(\frac{1}{\csc \left(x\right)}\right)^2}:$

$\csc ^2\left(x\right) verdadero$

5

Explicación paso a paso: