$\"\"$

\

The length of one arch of sine curve is $\"\"$ .

\

Arc length of the curve $\"\"$ in the interval $\"\"$ is $\"\"$ .

\

Consider the curve is $\"\"$ .

\

Apply derivative on each side with respect to $\"\"$ .

\

$\"\"$ .

\

Substitute $\"\"$ and $\"\"$ in $\"\"$ .

\

$\"\"$ .

\

$\"\"$

\

The length of one arch of cosine curve is $\"\"$ .

\

Arc length of the curve $\"\"$ in the interval $\"\"$ is $\"\"$ .

\

The curve is $\"\"$ .

\

Apply derivative on each side with respect to $\"\"$ .

\

$\"\"$ .

\

Substitute $\"\"$ and $\"\"$ in $\"\"$ .

\

$\"\"$ .

\

Substitute $\"\"$ .

\

$\"\"$

\

Consider $\"\"$ .

\

Apply derivative on each side with respect to $\"\"$ .

\

$\"\"$

\

If $\"\"$ then $\"\"$ .

\

If $\"\"$ then $\"\"$ .

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Substitute all the values in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

Therefore, the length of one arch of sine curve is equal to the length of one arch of cosine curve.

\

$\"\"$

\

The length of one arch of sine curve is equal to the length of one arch of cosine curve.