Question 942179
Verify the identity.
cotangent of {{{x}}} to the second power :{{{(ctg(x))^2}}}

divided by quantity cosecant of {{{x}}} plus one :{{{(csc(x)+1)}}}

so far we have {{{(ctg^2(x))/(csc(x)+1)}}}

then, that equals {{{(ctg^2(x))/(csc(x)+1)}}}=

to quantity one minus sine of {{{x}}}: {{{(1-sin(x))}}}

divided by sine of {{{x}}}: {{{(1-sin(x))/sin(x)}}}

so, we have:

{{{(ctg^2(x))/(csc(x)+1)=(1-sin(x))/sin(x)}}}  ..............since {{{1-sin(x) = cot^2(x)}}} and {{{cot^2(x)=(cos(x))^2/(sin(x))^2}}}, we have


{{{(ctg^2(x))/(csc(x)+1)=((cos(x))^2/(sin(x))^2)/(1/sin(x)+sin(x)/sin(x))}}}


or


{{{ctg^2(x)/(csc(x)+1)=((cos(x))^2/(sin(x))^cross(2))/((1+sin(x))/cross(sin(x))1)}}}


{{{ctg^2(x)/(csc(x)+1)=((cos(x))^2/sin(x))/(1+sin(x))}}}


{{{ctg^2(x)/(csc(x)+1)=(cos(x)highlight((cos(x)))/(highlight(sin(x))(1+sin(x))))}}}.............since {{{cos(x)/sin(x)=ctg(x)}}}



{{{ctg^2(x)/(csc(x)+1)=(cos(x) cot(x))/(sin(x)+1)}}} ................since{{{1+sin(x) = 1+1/(csc(x))}}}


{{{ctg^2(x)/(csc(x)+1)=(cos(x) cot(x))/(1+1/(csc(x)))}}}


{{{ctg^2(x)/(csc(x)+1)=(cos(x) cot(x))/((csc(x)+1)/csc(x)))}}}


{{{ctg^2(x)/(csc(x)+1)=(csc(x)cos(x) cot(x))/((csc(x)+1)))}}}  ............{{{csc(x)cos(x)=ctg(x)}}}


{{{ctg ^2 (x)/(csc(x)+1)=(cot(x) cot(x))/((csc(x)+1)))}}}


{{{ctg^2 (x)/(csc(x)+1)=cot^2(x)/((csc(x)+1)))}}}