University Calculus: Early Transcendentals (3rd Edition)

$\frac{dy}{d{\theta}}=-1$
Given that: $y=\csc^{-1}({sec\theta})$ differentiate with respect to $\theta$: $\frac{dy}{d{\theta}}=\frac{\frac{d\sec{\theta}}{d{\theta}}}{|sec{\theta}|{\sqrt{\sec^2{\theta}-1}}}$ $\frac{dy}{d{\theta}}=\frac{{-\tan{\theta}\sec{\theta}}}{|sec{\theta}|{\sqrt{\sec^2{\theta}-1}}}$ $\frac{dy}{d{\theta}}=\frac{{-\tan{\theta}}}{\tan{\theta}}$ $\frac{dy}{d{\theta}}=-1$