\(\displaystyle{\tan{{\left(\theta-\frac{\pi}{{4}}\right)}}}={\left({\tan{\theta}}-\frac{{\tan{{\left(\frac{\pi}{{4}}\right)}}}}{{{1}+{\tan{\theta}}\frac{{\tan{\pi}}}{{4}}}}=\right.}\)

\(\displaystyle\frac{{{\tan{\theta}}-{1}}}{{{1}+{\tan{\theta}}}}.\)

In a right triangle with hypotenuse length 13 and a leg length of 5, the other leg has a length \(\displaystyle\sqrt{{{13}^{{2}}-{5}^{{2}}}}=\sqrt{{144}}={12}\). In this triangle, when \(\displaystyle{\sin{\theta}}=\frac{{5}}{{13}}\) (opp/hyp) then \(\displaystyle{\tan{\theta}}={o}{p}\frac{{p}}{{a}}{d}{j}=\frac{{5}}{{12}}\).

Therefore \(\displaystyle{\tan{{\left(\theta-\frac{\pi}{{4}}\right)}}}=\frac{{\frac{{5}}{{12}}-{1}}}{{{1}+\frac{{5}}{{12}}}}=\frac{{-\frac{{7}}{{12}}}}{{\frac{{17}}{{12}}}}=-\frac{{7}}{{17}}.\)

\(\displaystyle\frac{{{\tan{\theta}}-{1}}}{{{1}+{\tan{\theta}}}}.\)

In a right triangle with hypotenuse length 13 and a leg length of 5, the other leg has a length \(\displaystyle\sqrt{{{13}^{{2}}-{5}^{{2}}}}=\sqrt{{144}}={12}\). In this triangle, when \(\displaystyle{\sin{\theta}}=\frac{{5}}{{13}}\) (opp/hyp) then \(\displaystyle{\tan{\theta}}={o}{p}\frac{{p}}{{a}}{d}{j}=\frac{{5}}{{12}}\).

Therefore \(\displaystyle{\tan{{\left(\theta-\frac{\pi}{{4}}\right)}}}=\frac{{\frac{{5}}{{12}}-{1}}}{{{1}+\frac{{5}}{{12}}}}=\frac{{-\frac{{7}}{{12}}}}{{\frac{{17}}{{12}}}}=-\frac{{7}}{{17}}.\)