Find the exact value of the following function without a calculator. csc((pi )/(4)) csc((pi )/(4))= square (Simplify your answer including any radicals. Use integers or fractions for any numbers in the expression.)

To find the exact value of \( \csc\left(\frac{\pi}{4}\right) \), we need to understand the relationship between cosecant and sine.<br /><br />The cosecant function is defined as:<br />\[ \csc(x) = \frac{1}{\sin(x)} \]<br /><br />First, let's find the value of \( \sin\left(\frac{\pi}{4}\right) \). The angle \( \frac{\pi}{4} \) radians is equivalent to 45 degrees. For a 45-degree angle in a right triangle, the sides are in the ratio \( 1:1:\sqrt{2} \). Therefore:<br />\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]<br /><br />Now, using the definition of cosecant:<br />\[ \csc\left(\frac{\pi}{4}\right) = \frac{1}{\sin\left(\frac{\pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]<br /><br />Thus, the exact value of \( \csc\left(\frac{\pi}{4}\right) \) is:<br />\[ \csc\left(\frac{\pi}{4}\right) = \sqrt{2} \]