Simplify (1-sin^2(t))/(sin^2)(t) to an expression involving a single squared trig function with no __

To simplify the expression \(\frac{1 - \sin^2(t)}{\sin^2(t)}\), we can use the Pythagorean identity for sine and cosine. The identity states:<br /><br />\[<br />\sin^2(t) + \cos^2(t) = 1<br />\]<br /><br />From this identity, we can express \(1 - \sin^2(t)\) in terms of \(\cos^2(t)\):<br /><br />\[<br />1 - \sin^2(t) = \cos^2(t)<br />\]<br /><br />Now, substitute \(\cos^2(t)\) into the original expression:<br /><br />\[<br />\frac{1 - \sin^2(t)}{\sin^2(t)} = \frac{\cos^2(t)}{\sin^2(t)}<br />\]<br /><br />The resulting expression is:<br /><br />\[<br />\frac{\cos^2(t)}{\sin^2(t)}<br />\]<br /><br />This can be written as:<br /><br />\[<br />\cot^2(t)<br />\]<br /><br />where \(\cot(t)\) is the cotangent function, which is the reciprocal of the tangent function.<br /><br />Thus, the simplified form of the given expression is:<br /><br />\[<br />\boxed{\cot^2(t)}<br />\]