Chapter 3 - Section 3.8 - Derivatives of Inverse Functions and Logarithms - Exercises - Page 175: 55

$$y'=-2\tan\theta$$

$$y=\ln(\cos^2\theta)$$ The derivative of $y$ is $$y'=\Big(\ln(\cos^2\theta)\Big)'=\frac{1}{\cos^2\theta}\times(\cos^2\theta)'$$ $$y'=\frac{1}{\cos^2\theta}\times2\cos\theta\times(\cos\theta)'=\frac{2}{\cos\theta}\times(-\sin\theta)$$ $$y'=-\frac{2\sin\theta}{\cos\theta}=-2\tan\theta$$