Question 1154166


{{{sec( 660) + cot (405)}}}..........express {{{sec( 660)}}}  with sin and cos


={{{1/cos( 660) + cot(405)}}}..use identity: {{{cos(x)=sin(90-x)}}}..since your {{{x=660}}}



={{{1/sin(90-660) + cot(405)}}}


{{{1/sin(-570) + cot(405)}}}...........use the following property : {{{sin (-x t)=- sin (x )}}}


={{{-1/sin(570) + cot(405)}}}


rewrite {{{sin(570)}}} .......... apply the periodicity of sin : {{{sin(570)=sin(x+360*k)=sin(x)}}}


so, {{{sin(570) =sin(210) }}}



={{{-1/sin(210) + cot(405)}}}................rewrite the angles for {{{cot(405=360+45)}}}={{{cot(45)}}}



={{{-1/sin(210) + cot(45)}}}......write {{{sin (210 )}}} as {{{sin (180 +30 )}}}



using the summation identity : {{{sin (x+y )= sin  (x ) cos (y )+ cos (x ) sin (y )}}}


so you have {{{sin (180 +30 )= sin  (180 ) cos (30 )+ cos (180 ) sin (30 )}}}


now use the following trivial identities:

{{{sin  (180 )=0}}}

{{{cos (180 )=-1}}}

{{{sin (30 )=1/2}}}

{{{cos (30 )=sqrt(3)/2}}}

then you have {{{sin (180 +30 )= 0*  (sqrt(3)/2 )+ (-1 ) (1/2 )=-1/2}}}

and {{{cot(45)=cos(45)/sin(45)=(1/sqrt(2)) /(1/sqrt(2))=1 }}}


so, you have:


={{{-1/(-1/2) + 1}}}

={{{2 + 1}}}

={{{3}}}

=> {{{sec( 660) + cot (405)=3}}}