Iven \( \cot \theta=12 / 5 \) - Ivaluats \( \frac{\sin ^{2} \theta}{\cos ^{2} \theta} \)

Let's break this down! Since \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), we can say that \( \cot \theta = \frac{5}{12} \) because it is the reciprocal of \( \frac{12}{5} \). This means that the adjacent side is 5 and the opposite side is 12 in a right triangle. By the Pythagorean theorem, the hypotenuse is \( \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \). Now, \( \sin \theta = \frac{opposite}{hypotenuse} = \frac{12}{13} \) and \( \cos \theta = \frac{adjacent}{hypotenuse} = \frac{5}{13} \). Finally, \( \frac{\sin^2 \theta}{\cos^2 \theta} = \left(\frac{12}{13}\right)^2 \div \left(\frac{5}{13}\right)^2 = \frac{144/169}{25/169} = \frac{144}{25} \). So, the value of \( \frac{\sin^2 \theta}{\cos^2 \theta} \) is \( \frac{144}{25} \).