15. Verify the identity as true or false. \( \cot (-\theta)=-\cot \theta \)

The given identity can be verified using the definition of the cotangent function. Cotangent is the ratio of the cosine to sine: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). For \( \cot(-\theta) \), we get \( \cot(-\theta) = \frac{\cos(-\theta)}{\sin(-\theta)} = \frac{\cos \theta}{-\sin \theta} = -\cot \theta \). Thus, as you've suggested, \( \cot(-\theta) = -\cot \theta \) is indeed true! Let’s tap into some more fascinating insights! The cotangent function is linked closely to various symmetrical properties in trigonometric functions. Analyzing how cotangent behaves under a negative angle highlights the odd function nature of cotangent, which leads to some neat properties in trigonometric identities! Furthermore, this identity pops up frequently in calculus, especially in solving integrals involving trigonometric functions or in physics problems when analyzing periodic motion or waves. Knowing these identities can simplify your calculations and deepen your understanding of the underlying concepts!