1. Start with the right-hand side: $\frac{\cot \theta}{\cos \theta} = \frac{\frac{\cos \theta}{\sin \theta}}{\cos \theta} = \frac{\cos \theta}{\sin \theta} \cdot \frac{1}{\cos \theta} = \frac{1}{\sin \theta} = \csc \theta$. Therefore, $\csc \theta = \frac{\cot \theta}{\cos \theta}$.
2. Use the Pythagorean identity: $\frac{1}{\sec^2 x} + \frac{1}{\csc^2 x} = \cos^2 x + \sin^2 x = 1$.
3. Factor the left-hand side: $\csc^2 y \tan^2 y - 1 = \left(\frac{1}{\sin^2 y}\right)\left(\frac{\sin^2 y}{\cos^2 y}\right) - 1 = \frac{1}{\cos^2 y} - 1 = \sec^2 y - 1 = \tan^2 y$.
4. Simplify each term: $\frac{\sec \theta}{\cos \theta} = \frac{1}{\cos^2 \theta}$ and $\frac{\tan \theta}{\cot \theta} = \frac{\frac{\sin \theta}{\cos \theta}}{\frac{\cos \theta}{\sin \theta}} = \frac{\sin^2 \theta}{\cos^2 \theta}$. Then, $\frac{1}{\cos^2 \theta} - \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{1 - \sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta}{\cos^2 \theta} = 1$.
5. Factor the left-hand side as a difference of squares: $\csc^4 x - \cot^4 x = (\csc^2 x - \cot^2 x)(\csc^2 x + \cot^2 x)$. Since $\csc^2 x - \cot^2 x = 1$, this simplifies to $1 \cdot (\csc^2 x + \cot^2 x) = \csc^2 x + \cot^2 x$.
6. Rearrange the equation: $\sec^4 y - \tan^4 y = (\sec^2 y - \tan^2 y)(\sec^2 y + \tan^2 y)$. Since $\sec^2 y - \tan^2 y = 1$, this becomes $1 \cdot (\sec^2 y + \tan^2 y) = \sec^2 y + \tan^2 y$.
7. Expand the left-hand side: $(1 - \tan \theta)^2 = 1 - 2\tan \theta + \tan^2 \theta$. Use the identity $1 + \tan^2 \theta = \sec^2 \theta$, so $1 + \tan^2 \theta - 2\tan \theta = \sec^2 \theta - 2\tan \theta$.
8. Use identities: $1 - \sin^2 x = \cos^2 x$ and $1 + \tan^2 x = \sec^2 x$. So, $(\cos^2 x)(\sec^2 x) = \cos^2 x \cdot \frac{1}{\cos^2 x} = 1$.
Parent Tip: Review the logic above to help your child master the concept of verifying trigonometric identities worksheet.
