VERIFICAR LAS IDENTIDADES TRIGONOMÉTRICAS EJERCICIOS RESUELTOS

VERIFICAR LAS SIGUIENTES IDENTIDADES.

1) $sen\, \theta \, sec\, \theta =tg\, \theta$
    $sen\, \theta \cdot \frac{1}{cos\, \theta }=tg\, \theta$
    $\frac{sen\, \theta}{cos\, \theta }=tg\, \theta$
    $tg\, \theta =tg\, \theta$

2) $(1-sen^{2}A)(1+tg^{2}A)=1$
    $(cos^{2}A)(sec^{2}A)=1$
    $\left ( cos^{2}A \right )\left ( \frac{1}{cos^{2}A} \right )=1$
                           $1=1$

3) $\left ( 1-cos\, \theta  \right )\left ( 1+sec\, \theta  \right )ctg\, \theta =sen\, \theta$
    $\left ( 1-cos\, \theta  \right )\left ( 1+\frac{1}{cos\, \theta } \right )\frac{cos\, \theta }{sen\, \theta }=sen\, \theta$
    $\left ( 1-cos\, \theta  \right )\left ( \frac{cos\, \theta +1}{cos\, \theta } \right )\frac{cos\, \theta }{sen\, \theta }=sen\, \theta$
    $\left ( 1-cos\, \theta  \right )\left ( cos\, \theta +1 \right )\frac{1}{sen\, \theta }=sen\, \theta$
    $\left ( 1-cos\, \theta  \right )\left ( 1+ cos\, \theta  \right )\frac{1}{sen\, \theta }=sen\, \theta$
    $\left ( 1-cos^{2}\, \theta  \right )\frac{1}{sen\, \theta }=sen\, \theta$
    $\left ( sen^{2}\, \theta  \right )\frac{1}{sen\, \theta }=sen\, \theta$
    $sen\, \theta=sen\, \theta$

4) $csc^{2}\, x\left ( 1-cos^{2}\, x \right )=1$
    $\frac{1}{sen^{2}\, x}\left ( sen^{2}\, x \right )=1$
    $1=1$

5) $\frac{sen\, \theta }{csc\, \theta }+\frac{cos\, \theta }{sec\, \theta }=1$
    $\frac{sen\, \theta }{\frac{1}{sen\, \theta }}+\frac{cos\, \theta }{\frac{1}{cos\, \theta}}=1$
    $sen^{2}\, \theta +cos^{2}\, \theta =1$
    $1=1$

6) $\frac{1-2cos^{2}A}{senA\, cosA}=tgA-ctgA$
    $\frac{1-2cos^{2}A}{senA\, cosA}=\frac{senA}{cosA}-\frac{cosA}{senA}$
    $\frac{1-2cos^{2}A}{senA\, cosA}=\frac{sen^{2}A-cos^{2}A}{senA\, cosA}$
    $\frac{1-2cos^{2}A}{senA\, cosA}=\frac{(1-cos^{2}A)-cos^{2}A}{senA\, cosA}$
    $\frac{1-2cos^{2}A}{senA\, cosA}=\frac{1-cos^{2}A-cos^{2}A}{senA\, cosA}$
    $\frac{1-2cos^{2}A}{senA\, cosA}=\frac{1-2cos^{2}A}{senA\, cosA}$

7) $tg^{2}\, x\, csc^{2}\, x\, ctg^{2}\, x\, sen^{2 }\, x=1$
    $(tg^{2}\, x\,ctg^{2}\, x) (csc^{2}\, x\,  sen^{2 }\, x)=1$
    $(1)(1)=1$
    $1=1$

8) $senA\, cosA\left ( tagA+ctgA \right )=1$
    $senA\, cosA\left ( \frac{senA}{cosA}+\frac{cosA}{senA} \right )=1$
    $senA\, cosA\left ( \frac{sen^{2}A+cos^{2}A}{senA\, cosA} \right )=1$
    $senA\, cosA\left ( \frac{1}{senA\, cosA} \right )=1$
    $1=1$

9) $1-\frac{cos^{2}\, \theta }{1+sen\, \theta }=sen\, \theta$
    $\frac{1+sen\, \theta-cos^{2}\, \theta }{1+sen\, \theta }=sen\, \theta$
    $\frac{1+sen\, \theta-(1-sen^{2}\, \theta) }{1+sen\, \theta }=sen\, \theta$
    $\frac{1+sen\, \theta-1+sen^{2}\, \theta }{1+sen\, \theta }=sen\, \theta$
    $\frac{sen\, \theta+sen^{2}\, \theta }{1+sen\, \theta }=sen\, \theta$
    $\frac{sen\, \theta(1+sen\, \theta )}{1+sen\, \theta }=sen\, \theta$
    $sen\, \theta =sen\, \theta$

10) $\frac{1}{sec\, \theta +tg\, \theta }=sec\, \theta -tg\, \theta$
    $\frac{1}{\frac{1}{cos\, \theta }+\frac{sen\, \theta }{cos\, \theta } }=sec\, \theta -tg\, \theta$
    $\frac{1}{\frac{1+sen\, \theta }{cos\, \theta } }=sec\, \theta -tg\, \theta$
    $\frac{cos\, \theta }{1+sen\, \theta }=sec\, \theta -tg\, \theta$
    $\frac{cos\, \theta }{1+sen\, \theta }\cdot \frac{1-sen\, \theta}{1-sen\, \theta}=sec\, \theta -tg\, \theta$
    $\frac{cos\, \theta(1-sen\, \theta) }{1-sen^{2}\, \theta }=sec\, \theta -tg\, \theta$
    $\frac{cos\, \theta(1-sen\, \theta) }{cos^{2}\, \theta }=sec\, \theta -tg\, \theta$
    $\frac{1-sen\, \theta }{cos\, \theta }=sec\, \theta -tg\, \theta$
    $\frac{1}{cos\, \theta }-\frac{sen\, \theta }{cos\, \theta }=sec\, \theta -tg\, \theta$
    $sec\, \theta -tg\, \theta =sec\, \theta -tg\, \theta$

11) $\frac{1}{1-senA}+\frac{1}{1+senA}=2\, sec^{2}A$
    $\frac{1+senA+1-senA}{(1-senA)(1+senA)}=2\, sec^{2}A$
    $\frac{2}{(1-sen^{2}A)}=2\, sec^{2}A$
    $\frac{2}{1-(1-cos^{2}A)}=2\, sec^{2}A$
    $\frac{2}{1-1+cos^{2}A}=2\, sec^{2}A$
    $\frac{2}{cos^{2}A}=2\, sec^{2}A$
    $2\left (\frac{1}{cos^{2}A}  \right )=2\, sec^{2}A$
    $2\, sec^{2}A=2\, sec^{2}A$

12) $\frac{1-cos\, x}{1+cos\, x}=\frac{sec\, x-1}{sec\, x+1}$
    $\frac{1-cos\, x}{1+cos\, x}=\frac{\frac{1}{cos\, x}-1}{\frac{1}{cos\, x}+1}$
    $\frac{1-cos\, x}{1+cos\, x}=\frac{\frac{1-cos\, x}{cos\, x}}{\frac{1+cos\, x}{cos\, x}}$
    $\frac{1-cos\, x}{1+cos\, x}=\frac{1-cos\, x}{1+cos\, x}$

13) $tg\, \theta \, sen\, \theta +cos\, \theta =sec\, \theta$
    $\left (\frac{sen\, \theta }{cos\, \theta}  \right ) \, sen\, \theta +cos\, \theta =sec\, \theta$
    $\frac{sen^{2}\, \theta }{cos\, \theta}  +cos\, \theta =sec\, \theta$
    $\frac{sen^{2}\, \theta+cos^{2 }\, \theta  }{cos\, \theta}  =sec\, \theta$
    $\frac{1 }{cos\, \theta}  =sec\, \theta$
    $sec\, \theta  =sec\, \theta$

14) $tg\, \theta -csc\, \theta\, sec\, \theta \, (1-2cos^{2}\, \theta )  =ctg\, \theta$
    $\left (\frac{sen\, \theta}{cos\, \theta }  \right ) -\frac{1}{sen\theta }\, \frac{1}{cos\, \theta } \, (1-2cos^{2}\, \theta )  =ctg\, \theta$
    $\frac{sen\, \theta}{cos\, \theta } - \frac{1}{sen\, \theta \, cos\, \theta } +\frac{2cos^{2}\, \theta }{sen\, \theta \, cos\, \theta } =ctg\, \theta$
    $\frac{sen^{2}\, \theta -1+2cos^{2}\, \theta }{sen\, \theta \, cos\, \theta } =ctg\, \theta$
    $\frac{1-cos^{2}\, \theta -1+2cos^{2}\, \theta }{sen\, \theta \, cos\, \theta } =ctg\, \theta$
    $\frac{cos^{2}\, \theta }{sen\, \theta \, cos\, \theta } =ctg\, \theta$
    $\frac{cos\, \theta }{sen\, \theta } =ctg\, \theta$
    $ctg\, \theta =ctg\, \theta$

15) $\frac{sen\, \theta }{sen\, \theta+cos\, \theta}=\frac{sec\, \theta }{sec\, \theta+csc\, \theta}$
    $\frac{sen\, \theta }{sen\, \theta+cos\, \theta}=\frac{\frac{1}{cos\, \theta} }{\frac{1}{cos\, \theta}+\frac{1}{sen\, \theta}}$
    $\frac{sen\, \theta }{sen\, \theta+cos\, \theta}=\frac{\frac{1}{cos\, \theta }}{\frac{sen\, \theta +cos\, \theta }{sen\, \theta \, cos\, \theta }}$
    $\frac{sen\, \theta }{sen\, \theta+cos\, \theta}=\frac{sen\, \theta \, cos\, \theta }{cos\, \theta (sen\, \theta +cos\, \theta )}$
    $\frac{sen\, \theta }{sen\, \theta+cos\, \theta}=\frac{sen\, \theta }{sen\, \theta +cos\, \theta }$

16) $\frac{senx+tgx}{ctgx+cscx}=senx\, tgx$
    $\frac{senx+\frac{senx}{cosx}}{\frac{cosx}{senx}+\frac{1}{senx}}=senx\, tgx$
    $\frac{\frac{senx\, cosx+senx}{cosx}}{\frac{cosx+1}{senx}}=senx\, tgx$
    $\frac{\frac{senx(cosx+1)}{cosx}}{\frac{cosx+1}{senx}}=senx\, tgx$
    $\frac{senx(cosx+1)senx}{cosx(cosx+1)}=senx\, tgx$
    $\frac{senx\, senx}{cosx}=senx\, tgx$
    $senx\, \frac{senx}{cosx}=senx\, tgx$
    $senx\, tgx=senx\, tgx$

17) $\frac{secx+cscx}{tgx+ctgx}=senx+cosx$
    $\frac{\frac{1}{cosx}+\frac{1}{senx}}{\frac{senx}{cosx}+\frac{cosx}{senx}}=senx+cosx$
    $\frac{\frac{senx+cosx}{senx\, cosx}}{\frac{sen^{2}x+cos^{2}x}{senx\, cosx}}=senx+cosx$
    $\frac{senx\, cosx}{sen^{2}x\, cos^{2}x}=senx+cosx$
    $\frac{senx\, cosx}{1}=senx+cosx$
    $senx+cosx=senx+cosx$

18) $\frac{sen^{3}\, \theta +cos^{3}\, \theta }{sen\, \theta +cos\, \theta}=1-sen\, \theta \, cos\, \theta$
    $\frac{(sen\, \theta +cos\, \theta)(sen^{2}\, \theta -sen\, \theta \, cos\, \theta +cos^{2}\, \theta ) }{sen\, \theta +cos\, \theta}=1-sen\, \theta \, cos\, \theta$
    $sen^{2}\, \theta -sen\, \theta \, cos\, \theta +cos^{2}\, \theta=1-sen\, \theta \, cos\, \theta$
    $(sen^{2}\, \theta +cos^{2}\, \theta)-sen\, \theta \, cos\, \theta =1-sen\, \theta \, cos\, \theta$
    $1-sen\, \theta \, cos\, \theta  =1-sen\, \theta \, cos\, \theta$

19) $ctg\, \theta +\frac{sen\, \theta }{1+cos\, \theta }=csc\, \theta$
    $\frac{cos\, \theta}{sen\, \theta} +\frac{sen\, \theta }{1+cos\, \theta }=csc\, \theta$
    $\frac{cos\, \theta \, (1+cos\, \theta)+sen^{2}\, \theta }{sen\, \theta \, (1+cos\, \theta) }=csc\, \theta$
    $\frac{cos\, \theta \, +cos^{2}\, \theta+sen^{2}\, \theta }{sen\, \theta \, (1+cos\, \theta) }=csc\, \theta$
    $\frac{cos\, \theta \, +(cos^{2}\, \theta+sen^{2}\, \theta) }{sen\, \theta \, (1+cos\, \theta) }=csc\, \theta$
    $\frac{cos\, \theta \, +1 }{sen\, \theta \, (1+cos\, \theta) }=csc\, \theta$
    $\frac{1 }{sen\, \theta \,  }=csc\, \theta$
    $csc\, \theta =csc\, \theta$

20) $\frac{sen\, \theta \, cos\, \theta }{cos^{2}\, \theta -sen^{2}\, \theta }=\frac{tg\, \theta }{1-tg^{2}\, \theta }$
    $\frac{sen\, \theta \, cos\, \theta }{cos^{2}\, \theta -sen^{2}\, \theta }=\frac{\frac{sen\, \theta}{cos\, \theta} }{1-\frac{sen^{2}\, \theta}{cos^{2}\, \theta} }$
    $\frac{sen\, \theta \, cos\, \theta }{cos^{2}\, \theta -sen^{2}\, \theta }=\frac{\frac{sen\, \theta}{cos\, \theta} }{\frac{cos^{2}\, \theta \, -sen^{2}\, \theta}{cos^{2}\, \theta} }$
    $\frac{sen\, \theta \, cos\, \theta }{cos^{2}\, \theta -sen^{2}\, \theta }=\frac{sen\, \theta \, cos^{2}\, \theta }{cos\, \theta (cos^{2}\, \theta -sen^{2}\, \theta )}$
    $\frac{sen\, \theta \, cos\, \theta }{cos^{2}\, \theta -sen^{2}\, \theta }=\frac{sen\, \theta \, cos\, \theta }{cos^{2}\, \theta -sen^{2}\, \theta }$

21) $(tg\, x+tg\, y)(1-ctg\, x\, ctg\, y)+(ctg\, x+ctg\, y)(1-tg\, x\, tg\, y)=0$
    $tg\, x-tg\, x\, ctg\, x\, ctg\, y+tg\, y-ctg\, x\, ctg\, y\, tg\, y+$
    $ctg\, x-ctg\, x\, tg\, xtg\, y+ctg\, y-tg\, xtg\, y\, ctg\, y=0$
    $tg\, x-(tg\, x\, ctg\, x)\, ctg\, y+tg\, y-ctg\, x\, (ctg\, y\, tg\, y)+$
    $ctg\, x-(ctg\, x\, tg\, x)tg\, y+ctg\, y-tg\, x\, (tg\, y\, ctg\, y)=0$
    $tg\, x-(1)\, ctg\, y+tg\, y-ctg\, x\, (1)+ctg\, x-(1)tg\, y+ctg\, y-tg\, x\, (1)=0$
    $tg\, x-\, ctg\, y+tg\, y-ctg\, x\, +ctg\, x-tg\, y+ctg\, y-tg\, x\, =0$
    eliminando términos semejantes
    $0=0$

22) $(x\,sen \theta -y\, cos \theta  )^{2}+(x\, cos\theta +y\, sen\theta )^{2}=r^{2}+y^{2}$
    $x^{2}\,sen^{2} \theta -2xy\, sen \theta\, cos \theta+ y^{2}cos^{2} \theta  +x^{2}\, cos^{2}\theta +2xy\, sen\theta \, cos\theta+$
    $y^{2}sen^{2}\theta =r^{2}+y^{2}$
    $x^{2}\,sen^{2} \theta+y^{2}sen^{2}\theta+y^{2}cos^{2} \theta+x^{2}cos^{2} \theta=r^{2}+y^{2}$
    $sen^{2} \theta(x^{2}+y^{2})+cos^{2} \theta(x^{2}+y^{2})=r^{2}+y^{2}$
    $(x^{2}+y^{2})(sen^{2} \theta+cos^{2} \theta)=r^{2}+y^{2}$
    $(x^{2}+y^{2})(1)=r^{2}+y^{2}$
    $x^{2}+y^{2}=r^{2}+y^{2}$

23) $(r\, sen\theta\,  cos\phi  )^{2}+(r\, sen\theta\,  sen\phi )^{2}+(rcos\theta )^{2}=r^{2}$
    $r^{2}\, sen^{2}\theta\,  cos^{2}\phi+r^{2}\, sen^{2}\theta\,  sen^{2}\phi+r^{2}cos^{2}\theta =r^{2}$
    $(r^{2}\, sen^{2}\theta\,  cos^{2}\phi+r^{2}\, sen^{2}\theta\,  sen^{2}\phi)+r^{2}cos^{2}\theta =r^{2}$
    $r^{2}\, sen^{2}\theta\,(  cos^{2}\phi+  sen^{2}\phi)+r^{2}cos^{2}\theta =r^{2}$
    $r^{2}\, sen^{2}\theta\,(  1)+r^{2}cos^{2}\theta =r^{2}$
    $r^{2}\, sen^{2}\theta+r^{2}cos^{2}\theta =r^{2}$
    $r^{2}\,(sen^{2}\theta+cos^{2}\theta) =r^{2}$
    $r^{2}\,(1) =r^{2}$
    $r^{2} =r^{2}$