Questions: Find the exact value of the expression. Do not use a calculator. csc^2 (π/4)-5 csc^2 (π/4)-5 =

Evaluate the expression \(\csc ^{2} \frac{\pi}{4}-5\).

Identify the value of \(\csc \frac{\pi}{4}\).

The cosecant function is the reciprocal of the sine function. Therefore, \(\csc \frac{\pi}{4} = \frac{1}{\sin \frac{\pi}{4}}\). Since \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), it follows that \(\csc \frac{\pi}{4} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}\).

Calculate \(\csc^2 \frac{\pi}{4}\).

Since \(\csc \frac{\pi}{4} = \sqrt{2}\), we have \(\csc^2 \frac{\pi}{4} = (\sqrt{2})^2 = 2\).

Subtract 5 from \(\csc^2 \frac{\pi}{4}\).

The expression becomes \(2 - 5 = -3\).

The exact value of the expression is \(\boxed{-3}\).

The exact value of the expression \(\csc ^{2} \frac{\pi}{4}-5\) is \(\boxed{-3}\).