Step 1

Given- \(\displaystyle{y}={1}–{3}{\tan{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}\)

To find- The equation of vertical asymptotes.

Concept Used- To find the vertical asymptotes of the function \(\displaystyle{y}={\frac{{{p}{\left({x}\right)}}}{{{q}{\left({x}\right)}}}}\), equate q(x) with 0 and solve accordingly.

Step 2

Explanation- Rewrite the given expression,

\(\displaystyle{y}={1}-{3}{\tan{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}\)

Simplify the above expression, we get,

\(\displaystyle{y}={1}-{\frac{{{3}}}{{{\cot{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}}}}\)

\(\displaystyle{y}={\frac{{{\cot{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}-{3}}}{{{\cot{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}}}}\)

For vertical asymptotes, equate denominator with zero, we get,

\(\displaystyle{\cot{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}={0}\)

\(\displaystyle{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}={0}\)

\(\displaystyle{2}{x}-\pi={0}\)

\(\displaystyle{2}{x}=\pi\)

\(\displaystyle{x}={\frac{{\pi}}{{{2}}}}\)

So, the equation of the vertical asymptotes is \(\displaystyle{x}={\frac{{\pi}}{{{2}}}}\).

Answer- Hence, the equation of the vertical asymptotes is \(\displaystyle{x}={\frac{{\pi}}{{{2}}}}\).

Given- \(\displaystyle{y}={1}–{3}{\tan{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}\)

To find- The equation of vertical asymptotes.

Concept Used- To find the vertical asymptotes of the function \(\displaystyle{y}={\frac{{{p}{\left({x}\right)}}}{{{q}{\left({x}\right)}}}}\), equate q(x) with 0 and solve accordingly.

Step 2

Explanation- Rewrite the given expression,

\(\displaystyle{y}={1}-{3}{\tan{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}\)

Simplify the above expression, we get,

\(\displaystyle{y}={1}-{\frac{{{3}}}{{{\cot{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}}}}\)

\(\displaystyle{y}={\frac{{{\cot{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}-{3}}}{{{\cot{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}}}}\)

For vertical asymptotes, equate denominator with zero, we get,

\(\displaystyle{\cot{{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}}}={0}\)

\(\displaystyle{\left({\frac{{{2}{x}-\pi}}{{{4}}}}\right)}={0}\)

\(\displaystyle{2}{x}-\pi={0}\)

\(\displaystyle{2}{x}=\pi\)

\(\displaystyle{x}={\frac{{\pi}}{{{2}}}}\)

So, the equation of the vertical asymptotes is \(\displaystyle{x}={\frac{{\pi}}{{{2}}}}\).

Answer- Hence, the equation of the vertical asymptotes is \(\displaystyle{x}={\frac{{\pi}}{{{2}}}}\).