Question

simplify \\( \frac{\sin ^{2}(t)+\cos ^{2}(t)}{\sin ^{2}(t)} \\) to an expression involving a single trig function with no fractions.

Explanation:

Step1: Use the Pythagorean identity

We know that \(\sin^{2}(t)+\cos^{2}(t) = 1\). So the expression becomes \(\frac{1}{\sin^{2}(t)}\).

Step2: Use the reciprocal identity

Since \(\csc(t)=\frac{1}{\sin(t)}\), then \(\frac{1}{\sin^{2}(t)}=\csc^{2}(t)\).

Answer:

\(\csc^{2}(t)\)