Question 291221
{{{1/(sec(theta)+tan(theta))-sec(theta)=tan(theta)}}} Start with the given equation.



{{{1/(1/cos(theta)+sin(theta)/cos(theta))-1/cos(theta)=tan(theta)}}} Rewrite {{{sec(theta)}}} as {{{1/cos(theta)}}}. Rewrite {{{tan(theta)}}} as {{{sin(theta)/cos(theta)}}}



{{{1/((1+sin(theta))/cos(theta))-1/cos(theta)=tan(theta)}}} Combine the inner fractions.



{{{cos(theta)/(1+sin(theta))-1/cos(theta)=tan(theta)}}} Multiply by the reciprocal.



{{{(cos(theta))^2/(cos(theta)(1+sin(theta)))-1/cos(theta)=tan(theta)}}} Multiply the first fraction by {{{cos(theta)/cos(theta)}}}



{{{(cos(theta))^2/(cos(theta)(1+sin(theta)))-(1+sin(theta))/(cos(theta)(1+sin(theta)))=tan(theta)}}} Multiply the second fraction by {{{(1+sin(theta))/(1+sin(theta))}}}




{{{((cos(theta))^2-(1+sin(theta)))/(cos(theta)(1+sin(theta)))=tan(theta)}}} Combine the fractions.



{{{((cos(theta))^2-1-sin(theta))/(cos(theta)(1+sin(theta)))=tan(theta)}}} Distribute



{{{(-1+(cos(theta))^2-sin(theta))/(cos(theta)(1+sin(theta)))=tan(theta)}}} Rearrange the terms.



{{{(-(1-(cos(theta))^2)-sin(theta))/(cos(theta)(1+sin(theta)))=tan(theta)}}} Factor out a negative 1 from the first two terms in the numerator.



{{{(-(sin(theta))^2-sin(theta))/(cos(theta)(1+sin(theta)))=tan(theta)}}} Replace {{{1-(cos(theta))^2}}} with {{{(sin(theta))^2}}}



{{{(-sin(theta)(sin(theta)+1))/(cos(theta)(1+sin(theta)))=tan(theta)}}} Factor out the GCF {{{-sin(theta)}}} from the numerator.



{{{(-sin(theta)(1+sin(theta)))/(cos(theta)(1+sin(theta)))=tan(theta)}}} Rearrange the terms.



{{{(-sin(theta)*highlight((1+sin(theta))))/(cos(theta)*highlight((1+sin(theta))))=tan(theta)}}} Highlight the common terms.



{{{(-sin(theta)*cross((1+sin(theta))))/(cos(theta)*cross((1+sin(theta))))=tan(theta)}}} Cancel out the common terms.



{{{-sin(theta)/cos(theta)=tan(theta)}}} Simplify 



{{{-tan(theta)=tan(theta)}}} Use the identity {{{tan(theta)=sin(theta)/cos(theta)}}}




So this show us that {{{1/(sec(theta)+tan(theta))-sec(theta)=-tan(theta)}}} for all values of {{{theta}}} in which are defined on {{{tan(theta)}}}. So either you left off the negative sign, or there's a typo somewhere. Double check the problem.