Question 1091318
Please use parenthesis to make your question clear.   You must have meant:

{{{ matrix(1,5, 1/(1+cos(x)), "+" ,1/(1-cos(x)), "=", 2/(sin^2(x)) ) }}}
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On left, multiply first term by (1-cos(x))/(1-cos(x)), 2nd term by (1+cos(x))/(1+cos(x)) :
{{{ 1/(1+cos(x)) * (1-cos(x))/(1-cos(x))  +  1/(1-cos(x)) * (1+cos(x))/(1+cos(x)) = 2/(sin^2(x))  }}}
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On left, multiply out numerator and denominator (its a common denominator due to the chosen multiplications):
{{{  (1-cos(x))/(1-cos^2(x)) + (1+cos(x))/(1-cos^2(x)) = 2/(sin^2(x)) }}}
simplifies to...
{{{      2 / (1-cos^2(x))  = 2/sin^2(x)  }}}

The final step is to recognize that {{{  1-cos^2(x) = sin^2(x) }}} and you are done.