Question 1191357

 {{{tan(x) - csc(x) = 0}}}

express with {{{sin(x) }}}and {{{cos(x)}}}


{{{sin(x)/cos(x) - 1/sin(x) = 0}}}


{{{(sin^2(x) - cos(x))/(sin(x)cos(x)) = 0}}}


use the rule: {{{ f (x )/g (x )=0}}} => only if  {{{f(x)=0}}}


{{{sin^2(x) - cos(x)= 0}}}..........use identity {{{sin^2(x)=1-cos^2(x)}}}


{{{1-cos^2(x)- cos(x)= 0 }}}


{{{-cos^2(x)- cos(x)+1= 0}}} ...........use quadratic formula


{{{cos(x)=(-(-1)+-sqrt((-1)^2-4(-1)*1))/(2(-1))}}}


{{{cos(x)=(1+-sqrt(1+4))/(-2)}}}


{{{cos(x)=-(1+-sqrt(5))/2}}}


solutions: {{{cos(x)=-(1+sqrt(5))/2}}} or {{{cos(x)=-(1-sqrt(5))/2}}}


{{{cos(x)=-(1+sqrt(5))/2}}} =>{{{x=cos^-1(-(1+sqrt(5))/2)}}} has{{{ no }}}solution


{{{cos(x)=-(1-sqrt(5))/2}}}

{{{cos(x)=(sqrt(5)-1)/2}}}

solutions:

{{{x=cos^-1((sqrt(5)-1)/2)+2pi*n}}}
or
{{{x=2pi-cos^-1((sqrt(5)-1)/2)+2pi*n}}}


 solutions in decimal form:

in radians

{{{x=0.90455  +2pi* n}}}
{{{x=2pi -0.90455+2pi* n}}}


in degrees

{{{x=51.82729}}}°+{{{360}}}°*{{{n}}}
{{{x=308.17}}}°+{{{360}}}°*{{{n}}}