Question 1187011

1.
{{{( csc(theta)*cos(theta))/cot(theta)=1}}}

manipulate left side...........use identity{{{ csc(theta)=1/sin(theta)}}}

{{{( (1/sin(theta))*cos(theta))/cot(theta)}}}

={{{( cos(theta)/sin(theta))/cot(theta)}}}................use identity {{{cos(theta)/sin(theta)=cot(theta)}}}

={{{cot(theta)/cot(theta)}}}

={{{1}}}



2,  

 {{{(sin(theta)-cos(theta))^2=1-sin(2theta)}}}

manipulate left side

{{{ (sin(theta)-cos(theta))^2}}}......square
 
={{{sin^2(theta)-2sin(theta)cos(theta)+cos^2(theta)}}}...........use identity {{{cos^2(theta)=1-sin^2(theta)}}}

={{{sin^2(theta)-2sin(theta)cos(theta)+1-sin^2(theta)}}}..........simplify

={{{-2sin(theta)cos(theta)+1}}}.........rearange

={{{1-2sin(theta)cos(theta) }}}..........................use double angle identity {{{2sin(theta)cos(theta)=sin(2theta)}}}

={{{1-sin(2theta)}}}