Question 1202473
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I'll keep the left hand side (LHS) the same, while altering the right hand side (RHS)
The goal is to make the LHS and RHS expressions to be identical to each other.
{{{(tan(x))/(1+tan(x)) = 1/(1+cot(x))}}}


{{{(tan(x))/(1+tan(x)) = 1/(1+1/tan(x))}}} Rewrite cot(x) as 1/tan


{{{(tan(x))/(1+tan(x)) = ((tan(x))/(tan(x)))*(1/(1+1/tan(x)))}}} Multiply top and bottom by tan/tan, which is equivalent to 1.


{{{(tan(x))/(1+tan(x)) = (tan(x))/(tan(x)*(1+1/tan(x)))}}}


{{{(tan(x))/(1+tan(x)) = (tan(x))/(tan(x)+tan(x)*(1/tan(x)))}}} Distribute in the denominator


{{{(tan(x))/(1+tan(x)) = (tan(x))/(tan(x)+1)}}}


{{{(tan(x))/(1+tan(x)) = (tan(x))/(1+tan(x))}}}


The identity has been confirmed.


If you wanted to alter the LHS, then keep the RHS the same. You could divide each piece of the LHS by tan(x) to effectively reverse the process shown above.
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