# "We can analyze as follows:" #
# \qquad \qquad \qquad sin[ cot^{-1}( 1 + x ) ] \ = \ cos[ tan^{-1}( x ) ]. #
# "Let:" \qquad \qquad \qquad A = cot^{-1}( 1 + x ) \qquad "and" \qquad B = tan^{-1}( x ). #
# "Then we have:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad sin[ A ] \ = \ cos[ B ]. #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\quad 1/csc[ A ] \ = \ 1/sec[ B ]. #
# \qquad \qquad \qquad \qquad \qquad \qquad \quad 1/sqrt{ 1 + cot^2 A } \ = \ 1/sqrt{ 1 + tan^2 A }. #
# \qquad 1/sqrt{ 1 + cot^2 [ cot^{-1}( 1 + x ) ] } \ = \ 1/sqrt{ 1 + tan^2 [ tan^{-1}( x ) ] }. #
# 1/sqrt{ 1 + ( cot[ cot^{-1}( 1 + x ) ] )^2 } \ = \ 1/sqrt{ 1 + ( tan[ tan^{-1}( x ) ] )^2 ]. #
# \qquad \qquad \qquad \qquad \qquad \quad 1/sqrt{ 1 + ( 1 + x )^2 } \ = \ 1/sqrt{ 1 + x^2 ]. #
# \qquad \qquad \qquad \qquad \qquad \quad \ sqrt{ 1 + ( 1 + x )^2 } \ = \ sqrt{ 1 + x^2 ]. #
# \qquad \qquad \qquad \qquad \qquad \qquad \quad \ 1 + ( 1 + x )^2 \ = \ 1 + x^2. #
# \qquad \qquad \qquad \qquad \ 1 + ( 1 + 2 x + x^2 ) \ = \ 1 + x^2. #
# \qquad \qquad \qquad \qquad \qquad \qquad \quad 2 + 2 x + color{red}cancel{ x^2 } \ = \ 1 + color{red}cancel{ x^2 }. #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ 2 + 2 x \ = \ 1 . #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \ 2 x \ = \ -1 . #
# qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \ \ x \ = \ -1/2 . #
# "This is our solution !!" #
# "So, summarizing:" #
# \qquad "The solution set of:" \qquad \quad sin[ cot^{-1}( 1 + x ) ] \ = \ cos[ tan^{-1}( x ) ]#
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ "is:" qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \ \ x \ = \ -1/2 \quad. \qquad \qquad \ square #
