Question 148632
Prove the identity:

{{{csc(x)+cot(x)=sin(x)/(1-cos(x))}}}
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Replace {{{csc(x)}}} on left side by {{{1/(sin(x))}}}
Replace {{{cot(x)}}} on left side by {{{(cos(x))/(sin(x))}}}

{{{1/(sin(x))+(cos(x))/(sin(x))}}}

Combine the numerators over the common denominator:

{{{(1+cos(x))/(sin(x))}}}

Form the conjugate of the numerator by changing the sign
of the second term:  {{{1-cos(x)}}}, put it over itself,
like this {{{(1-cos(x))/(1-cos(x))}}}, which equals 1,
so we can multiply by it without changing the value:

{{{((1+cos(x))/(sin(x)))((1-cos(x))/(1-cos(x)))}}}

Indicate the multiplications of the tops and bottoms:

{{{((1+cos(x))(1-cos(x)))/(sin(x)(1-cos(x))))}}}

FOIL out the top:

{{{ (  1-cos(x)+cos(x)-Cos^2x )/( sin(x)(1-cos(x))) }}}

Simplify by canceling in the top:

{{{ (  1-cross(cos(x))+cross(cos(x))-Cos^2x )/( sin(x)(1-cos(x))) }}}

{{{ (  1-Cos^2x )/( sin(x)(1-cos(x))) }}}

Replace {{{1-Cos^2x}}} by {{{Sin^2x}}}

{{{ (  Sin^2x )/( sin(x)(1-cos(x))) }}}

The {{{sin(x)}}} in the bottom cancels out the
square in the top:

{{{ (  Sin^cross(2)(x) )/( cross(sin(x))(1-cos(x))) }}}

{{{sin(x)/(1-cos(x))}}}

Edwin</pre>