Identidades Trigonométricas

  Geometria

En el estudio de las identidades es conveniente construir de acuerdo a procesos, de tal que las identidades notables sean un complemento adecuado que permita inducir estrategias de demostración en base a algo verdadero.

La identidad inicial que da sentido a este estudio es

$$sen^2(\alpha)+cos^2(\alpha)=1$$

lo cual se puede verificar en la figura , usando el teorema de Pitagoras.

Si operamos sobre ella ciertas condiciones, tendremos las otras dos

$$\frac{sen^2(\alpha)}{sen^2(\alpha)} + \frac{cos^2(\alpha)}{sen^2(\alpha)}\rightarrow\frac{1}{sen^2(\alpha)}=1+ctg^2(\alpha)=csc^2(\alpha)$$

$$\frac{sen^2(\alpha)}{cos^2(\alpha)}+\frac{cos^2(\alpha)}{cos^2(\alpha)}\rightarrow\frac{1}{cos^2(\alpha)}=tan^2(\alpha)+1=sec^2(\alpha)$$

 

y podemos comenzar a trabajar las ideas

1.- $ Cos(\alpha)Tan (\alpha)=sen(\alpha)$

$$cos(\alpha)tan(\alpha)=sen(\alpha)$$

$$cos(\alpha)\frac{sen(\alpha)}{cos(\alpha)}=sen(\alpha)$$

$$sen(\alpha)=sen(\alpha)$$

2.- $Sen(\alpha)Sec (\alpha)=Tan(\alpha)$

$$Sen(\alpha)\cdot Sec (\alpha)=Tan(\alpha)$$

$$Sen(\alpha)\cdot \frac{1}{cos \alpha} =Tan(\alpha)$$

$$Tan(\alpha) =Tan(\alpha)$$

 

3.- $sen (\alpha)\cdot ctg(\alpha)=cos(\alpha)$

$$sen (\alpha)\cdot ctg(\alpha)=cos(\alpha)$$

$sen (\alpha)\cdot \frac{cos(\alpha)}{sen(\alpha)}(\alpha)=cos(\alpha)$

$cos (\alpha)=cos(\alpha)$

4.- $sen(\alpha)tan(\alpha)+cos(\alpha)=sec(\alpha)$

$$sen(\alpha)tan(\alpha)+cos(\alpha)=sec(\alpha)$$

$$sen(\alpha)\frac{sen(\alpha}{cos()\alpha}+cos(\alpha)=sec(\alpha)$$

$$\frac{sen^2(\alpha)}{cos\alpha}+\frac{cos(\alpha}{1}=sec(\alpha)$$

$$\frac{sen^2(\alpha)+cos^(\alpha)}{cos(\alpha}=sec(\alpha)$$

$$\frac{1}{cos{\alpha}}=sec(\alpha)$$

$$sec(\alpha)=sec(\alpha)$$

5.- $csc(\alpha)-sen(\alpha)=ctg(\alpha)cos(\alpha)$

$$csc(\alpha)-sen(\alpha)=ctg(\alpha)cos(\alpha)$$

$$\frac{1}{sen (\alpha)}-\frac{sen(\alpha)}{1}=ctg(\alpha)cos(\alpha)$$

$$\frac{1-sen^2{\alpha}}{sen(\alpha}=ctg(\alpha)cos(\alpha)$$

$$\frac{cos^2(\alpha}{sen(\alpha}=ctg(\alpha)cos(\alpha)$$

$$\frac{\cos(\alpha)}{sen(\alpha)}\cdot cos(\alpha)=ctg(\alpha)cos(\alpha)$$

$$ctg(\alpha)cos(\alpha)=ctg(\alpha)cos(\alpha)$$

6.- $\sqrt{\frac{1-cos(\alpha)}{1+cos(\alpha)}}=csc(\alpha)-cot(\alpha)$

$$\sqrt{\frac{1-cos(\alpha)}{1+cos(\alpha)}}=csc(\alpha)-cot(\alpha)$$

$${\frac{1-cos(\alpha)}{1+cos(\alpha)}}=\left(csc(\alpha)-cot(\alpha)\right)^2$$

$${\frac{1-cos(\alpha)}{1+cos(\alpha)}}=csc^2(\alpha)-2csc(\alpha)cot(\alpha)+cot^2(\alpha)$$

$${\frac{1-cos(\alpha)}{1+cos(\alpha)}}=\frac{1}{sen^2(\alpha)}-2\frac{1}{sen(\alpha)}\frac{cos(\alpha)}{sen(\alpha)}+\frac{cos^2(\alpha)}{sen^2(\alpha)}$$

$${\frac{1-cos(\alpha)}{1+cos(\alpha)}}=\frac{1}{sen^2(\alpha)}-\frac{2cos(\alpha)}{sen^2(\alpha)}+\frac{cos^2(\alpha)}{sen^”(\alpha)}$$

$${\frac{1-cos(\alpha)}{1+cos(\alpha)}}=\frac{1-2cos(\alpha)+cos^2(\alpha)}{sen^2(\alpha)}$$

$${\frac{1-cos(\alpha)}{1+cos(\alpha)}}=\frac{\left(  1-cos(\alpha)     \right)^2}{1-cos^2(\alpha)}$$

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

$${\frac{1-cos(\alpha)}{1+cos(\alpha)}}={\frac{1-cos(\alpha)}{1+cos(\alpha)}}$$

 

7.-$\left( sen(\alpha)+cos(\alpha)\right)^2+\left( sen(\alpha)-cos(\alpha)\right)^2=2$$

 

8.- $\left(sen(\alpha)+csc(\alpha)\right)^2=sen^2(\alpha)+ctg^2(\alpha)+3$

$$sen^2(\alpha)+2sen(\alpha)csc(\alpha)+csc^2(\alpha)=sen^2(\alpha)+ctg^2(\alpha)+3$$

$$sen^2(\alpha)+2sen(\alpha)\frac{1}{sen(\alpha)}+csc^2(\alpha)=sen^2(\alpha)+ctg^2(\alpha)+3$$

$$sen^2(\alpha)+2+csc^2(\alpha)=sen^2(\alpha)+ctg^2(\alpha)+3$$

$$sen^2(\alpha)+2+1+ctg^2(\alpha)=sen^2(\alpha)+ctg^2(\alpha)+3$$

$$sen^2(\alpha)+3+ctg^2(\alpha)=sen^2(\alpha)+ctg^2(\alpha)+3$$

$$sen^2(\alpha)+ctg^2(\alpha)+3=sen^2(\alpha)+ctg^2(\alpha)+3$$

 

9.- $\frac{sen(\alpha)}{1+cos(\alpha)}+\frac{1+cos(\alpha)}{sen(\alpha)}=2csc(\alpha)$

$$\frac{sen(\alpha)}{1+cos(\alpha)}+\frac{1+cos(\alpha)}{sen(\alpha)}=2csc(\alpha)$$

$$\frac{sen^2(\alpha)+1+2cos(\alpha)+cos^2(\alpha)}{(1+cos(\alpha))(sen(\alpha)}=2csc(\alpha)$$

$$\frac{1+1+2cos(\alpha)}{(1+cos(\alpha))(sen(\alpha)}=2csc(\alpha)$$

$$\frac{2+2cos(\alpha)}{(1+cos(\alpha))(sen(\alpha))}=2csc(\alpha)$$

$$2\frac{1+cos(\alpha)}{(1+cos(\alpha))(sen(\alpha))}=2csc(\alpha)$$

$$\frac{2(1+cos(\alpha))}{(1+cos(\alpha))(sen(\alpha))}=2csc(\alpha)$$

$$\frac{2)}{sen(\alpha))}=2csc(\alpha)$$

$$2csc(\alpha)=2csc(\alpha)$$

10.- $\frac{csc(\alpha)}{ctg(\alpha)+tg(\alpha)}=cos(\alpha)$

$$\frac{csc(\alpha)}{ctg(\alpha)+tg(\alpha)}=cos(\alpha)$$

$$\frac{\frac{1}{sen(\alpha)}}{\frac{cos(\alpha)}{sen(\alpha)}+\frac{sen(\alpha)}{cos(\alpha)}}=cos(\alpha)$$

$$\frac{\frac{1}{sen(\alpha)}}{\frac{cos^2(\alpha)+sen^2(\alpha)}{sen(\alpha)cos(\alpha)}}=cos(\alpha)$$

$$\frac{\frac{1}{sen(\alpha)}}{\frac{1}{sen(\alpha)cos(\alpha)}}=cos(\alpha)$$

$$\frac{\frac{1}{sen(\alpha)}}{\frac{1}{sen(\alpha)cos(\alpha)}}=cos(\alpha)$$

$${\frac{1}{sen(\alpha)}}{\frac{sen(\alpha)cos(\alpha)}{1}}=cos(\alpha)$$

$$cos(\alpha)=cos(\alpha)$$

11.- $cos^4(\alpha)-sin^4(\alpha)+1=2cos^2(\alpha)$

$$cos^4(\alpha)-sin^4(\alpha)+1=2cos^2(\alpha)$$

$$(cos^2(\alpha)+sin^2(\alpha))(cos^2(\alpha)-sin^2(\alpha))+1=2cos^2(\alpha)$$

$$(1)(cos^2(\alpha)-sin^2(\alpha))+1=2cos^2(\alpha)$$

$$(cos^2(\alpha)-sin^2(\alpha))+1=2cos^2(\alpha)$$

$$cos^2(\alpha)-(1-cos^2(\alpha))+1=2cos^2(\alpha)$$

$$cos^2(\alpha)-1+cos^2(\alpha)+1=2cos^2(\alpha)$$

$$cos^2(\alpha)+cos^2(\alpha)=2cos^2(\alpha)$$

$$2cos^2(\alpha)=2cos^2(\alpha)$$

12.-  $sec^4(\alpha)-sec^2(\alpha)=tan^4(\alpha)+tan^2(\alpha)$

$$sec^4(\alpha)-sec^2(\alpha)=tan^4(\alpha)+tan^2(\alpha)$$

$$sec^2(\alpha)(1-sec^2(\alpha))=tan^4(\alpha)+tan^2(\alpha)$$

pero  $sec^2(\alpha)=tan^2(\alpha)+1$, entonces $sec^2(\alpha)-1=tan^2(\alpha)$

$$(tan^2(\alpha)+1)(tan^2(\alpha))=tan^4(\alpha)+tan^2(\alpha)$$

$$tan^2(\alpha)+tan^4(\alpha)=tan^4(\alpha)+tan^2(\alpha)$$

$$tan^4(\alpha)+tan^2(\alpha)=tan^4(\alpha)+tan^2(\alpha)$$

13.-  $\sqrt{\frac{tan^2(\alpha)}{1+tan^2(\alpha)}}=sen(\alpha)$

$$\sqrt{\frac{tan^2(\alpha)}{1+tan^2(\alpha)}}=sen(\alpha)$$

$$\sqrt{\frac{tan^2(\alpha)}{sec^2(\alpha)}}=sen(\alpha)$$

$$\sqrt{\frac{\frac{sen^2(\alpha)}{cos^2(\alpha)}}{\frac{1}{cos^2(\alpha)}}}=sen(\alpha)$$

$$\sqrt{{\frac{sen^2(\alpha)}{cos^2(\alpha)}}{\frac{cos^2(\alpha)}{1}}}=sen(\alpha)$$

$$\sqrt{sen^2(\alpha)}=sen(\alpha)$$

$$sen(\alpha)=sen(\alpha)$$

14.- $(sec(\alpha)+cos(\alpha))(sec(\alpha)-cos(\alpha))=tan^2(\alpha)+sen^2(\alpha)$

$$(sec(\alpha)+cos(\alpha))(sec(\alpha)-cos(\alpha))=tan^2(\alpha)+sen^2(\alpha)$$

$$sec^2(\alpha)-cos^2(\alpha)=tan^2(\alpha)+sen^2(\alpha)$$

$$tan^2(\alpha)+1-cos^2(\alpha))=tan^2(\alpha)+sen^2(\alpha)$$

$$tan^2(\alpha)+sen^2(\alpha)=tan^2(\alpha)+sen^2(\alpha)$$

15.-  $ctg^4(\alpha)+ctg^2(\alpha)=csc^4(\alpha)-csc^2(\alpha)$

$$ctg^4(\alpha)+ctg^2(\alpha)=csc^4(\alpha)-csc^2(\alpha)$$

$$ctg^2(\alpha)(ctg^2(\alpha)+1)=csc^4(\alpha)-csc^2(\alpha)$$

pero $ctg^2(\alpha)=csc^2(\alpha)-1$

$$(csc^2(\alpha)-1)((csc^2(\alpha)-1)+1)=csc^4(\alpha)-csc^2(\alpha)$$

$$(csc^2(\alpha)-1)(csc^2(\alpha))=csc^4(\alpha)-csc^2(\alpha)$$

$$csc^4(\alpha)-csc^2(\alpha)=csc^4(\alpha)-csc^2(\alpha)$$

16.- $(1+tan^2(\alpha))cos^2(\alpha)=1$

$$(1+tan^2(\alpha))cos^2(\alpha)=1$$

$$sec^2(\alpha))cos^2(\alpha)=1$$

$$\frac{1}{cos^2(\alpha)})cos^2(\alpha)=1$$

$$1=1$$

17.-  $sen^2(\alpha)+sen^2(\alpha)\cdot tan^2(\alpha)=tan^2(\alpha)$

$$sen^2(\alpha)+sen^2(\alpha)\cdot tan^2(\alpha)=tan^2(\alpha)$$

$$sen^2(\alpha)(1+tan^2(\alpha)=tan^2(\alpha)$$

$$sen^2(\alpha)(sec^2(\alpha))=tan^2(\alpha)$$

$$\frac{sen^2(\alpha)}{cos^2(\alpha)}=tan^2(\alpha)$$

$$tan^2(\alpha)=tan^2(\alpha)$$

18.-  $sec^2(\alpha)+csc^2(\alpha)=sec^2(\alpha)\cdot csc^2(\alpha)$

$$sec^2(\alpha)+csc^2(\alpha)=sec^2(\alpha)\cdot csc^2(\alpha)$$

$$\frac{1}{cos^2(\alpha)}+\frac{1}{sen^2(\alpha)}=sec^2(\alpha)\cdot csc^2(\alpha)$$

$$\frac{sen^2(\alpha)+cos^2(\alpha)}{cos^2(\alpha)sen^2(\alpha)}=sec^2(\alpha)\cdot csc^2(\alpha)$$

$$\frac{1}{cos^2(\alpha)sen^2(\alpha)}=sec^2(\alpha)\cdot csc^2(\alpha)$$

$$\frac{1}{cos^2(\alpha)}\frac{1}{sen^2(\alpha)}=sec^2(\alpha)\cdot csc^2(\alpha)$$

$$sec^2(\alpha)\cdot csc^2(\alpha)=sec^2(\alpha)\cdot csc^2(\alpha)$$

19.-  $tan(\alpha)+ctg(\alpha)=sec(\alpha)\cdot csc(\alpha)$

$$tan(\alpha)+ctg(\alpha)=sec(\alpha)\cdot csc(\alpha)$$

$$\frac{sen(\alpha)}{cos(\alpha)}+\frac{cos(\alpha)}{sen(\alpha)}=sec(\alpha)\cdot csc(\alpha)$$

$$\frac{sen^2(\alpha)+cos^2(\alpha)}{sen(\alpha)cos(\alpha)}=sec(\alpha)\cdot csc(\alpha) $$

$$\frac{1}{sen^2(\alpha)}\cdot{1}{cos(\alpha)}=sec(\alpha)\cdot csc(\alpha)$$

$$sec(\alpha)\cdot csc(\alpha)=sec(\alpha)\cdot csc(\alpha)$$

20.-  $(1+ctg^2(\alpha))sen^2(\alpha)=1$

$$(1+ctg^2(\alpha))sen^2(\alpha)=1$$

$$csc^2(\alpha)\cdot sen^2(\alpha)=1$$

$$\frac{1}{sen^2(\alpha)}\cdot sen^2(\alpha)=1$$

$$1=1$$

21.-  $cos^4(\alpha)-sen^4(\alpha)-2cos^2(\alpha)=-1$

$$cos^4(\alpha)-sen^4(\alpha)-2cos^2(\alpha)=-1$$

$$\left(cos^2(\alpha)-sen^2(\alpha)\right)\left(cos^2(\alpha)+sen^2(\alpha)\right)-2cos^2(\alpha)=-1$$

$$\left(cos^2(\alpha)-sen^2(\alpha)\right)\left(1\right)-2cos^2(\alpha)=-1$$

$$\left(cos^2(\alpha)-sen^2(\alpha)\right)-2cos^2(\alpha)=-1$$

$$\left(-cos^2(\alpha)-sen^2(\alpha)\right)=-1$$

$$-\left(cos^2(\alpha)+sen^2(\alpha)\right)=-1$$

$$-\left(1\right)=-1$$

22.-  $sen^3(\alpha)cos(\alpha)+cos^3(\alpha)sen(\alpha)=sen(\alpha)cos(\alpha)$

$$sen^3(\alpha)cos(\alpha)+cos^3(\alpha)sen(\alpha)=sen(\alpha)cos(\alpha)$$

$$sen(\alpha)cos(\alpha)\cdot \left(sen^2(\alpha)+cos^2(\alpha)\right)=$$

$$sen(\alpha)cos(\alpha)\left(1\right)=$$

$$sen(\alpha)cos(\alpha)=sen(\alpha)cos(\alpha)$$

23.-  $\frac{sen (\alpha)}{1+cos(\alpha)}+\frac{1+cos(\alpha)}{sen(\alpha)}=2csc(\alpha)$

$$\frac{sen (\alpha)}{1+cos(\alpha)}+\frac{1+cos(\alpha)}{sen(\alpha)}=2csc(\alpha)$$

$$\frac{sen^2(\alpha)+(1+cos(\alpha))^2}{(1+cos(\alpha))sen(\alpha)}=2csc(\alpha)$$

$$\frac{sen^2(\alpha)+1+2cos(\alpha)+cos^2(\alpha)}{(1+cos(\alpha))sen(\alpha)}=2csc(\alpha)$$

$$\frac{sen^2(\alpha)+cos^2(\alpha)+1+2cos(\alpha)}{(1+cos(\alpha))sen(\alpha)}=2csc(\alpha)$$

$$\frac{1+1+2cos(\alpha)}{(1+cos(\alpha))sen(\alpha)}=2csc(\alpha)$$

$$\frac{2+2cos(\alpha)}{(1+cos(\alpha))sen(\alpha)}=2csc(\alpha)$$

$$\frac{2(1+cos(\alpha))}{(1+cos(\alpha))sen(\alpha)}=2csc(\alpha)$$

$$\frac{2}{sen(\alpha)}=2csc(\alpha)$$

$$2csc(\alpha)=2csc(\alpha)$$

24.-  $ctg(\alpha)+\frac{sen(\alpha)}{1+cos(\alpha)}=csc(\alpha)$

$$ctg(\alpha)+\frac{sen(\alpha)}{1+cos(\alpha)}=csc(\alpha)$$

$$\frac{cos(\alpha)}{sen(\alpha)}+\frac{sen(\alpha)}{1+cos(\alpha)}=csc(\alpha)$$

$$\frac{cos(\alpha)+cos^2(\alpha)+sen^2(\alpha)}{sen(\alpha)(1+cos(\alpha))}=csc(\alpha)$$

$$\frac{cos(\alpha)+1}{sen(\alpha)(1+cos(\alpha))}=csc(\alpha)$$

$$\frac{1}{sen(\alpha)}=csc(\alpha)$$

$$csc(\alpha)=csc(\alpha)$$

25.-  $\sqrt{(1-sen(\alpha))(1+sen(\alpha))}=\frac{1}{sec(\alpha)}$

$$\sqrt{(1-sen(\alpha))(1+sen(\alpha))}=\frac{1}{sec(\alpha)}$$

$$\sqrt{(1-sen^2(\alpha))}=\frac{1}{sec(\alpha)}$$

$$\sqrt{cos^2(\alpha)}=\frac{1}{sec(\alpha)}$$

$${cos(\alpha)}=\frac{1}{sec(\alpha)}$$

$$\frac{1}{sec(\alpha)}=\frac{1}{sec(\alpha)}$$

26.-  $sen^(\alpha)cos^2(\alpha)+cos^4(\alpha)=cos^2(\alpha)$

$$sen^(\alpha)cos^2(\alpha)+cos^4(\alpha)=cos^2(\alpha)$$

$$cos^2(\alpha)\left(sen^2(\alpha)+cos^2(\alpha)\right)=cos^2(\alpha)$$

$$cos^2(\alpha)\left(1\right)=cos^2(\alpha)$$

$$cos^2(\alpha)=cos^2(\alpha)$$

27.-  $\frac{cos(\alpha)}{1-tan(\alpha)}+\frac{sen(\alpha)}{1-ctg(\alpha)}=sen(\alpha)+cos(\alpha)$

$$\frac{cos(\alpha)}{1-tan(\alpha)}+\frac{sen(\alpha)}{1-ctg(\alpha)}=sen(\alpha)+cos(\alpha)$$

 

 

 

(\alpha)

En proceso

 

 

 

 

 

 

 

 

 

 

 

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