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