Write the tangent expression as the reciprocal of the cotangent


 
which is the same as:



Since both numerator and denominator approach 0 as x approaches -1,
we can use L'Hopital's rule by taking deritatives of top and bottom:

[The software available here writes Csc²x as Csc²x and same
for other trig functions squared. Sorry! )



Write the cosecant squared in the denominator as a sine squared in the
numerator, and remember to keep the - sign:



Change division by  as multiplication by its reciprocal 



Now when x is close to 1 this is close to what we get when we
substitute 1 for x









Edwin