1. **Problem Statement:** Given the identity $$\frac{1 - \tan^2\theta}{\sec^2\theta} = \cos 2\theta,$$ (a) State a non-permissible value of $\theta$. (b) Prove the identity for all permissible values of $\theta$. 2. **Step (a): Find non-permissible values** - The expression involves $\tan\theta$ and $\sec\theta$. - $\tan\theta = \frac{\sin\theta}{\cos\theta}$ is undefined when $\cos\theta = 0$. - $\sec\theta = \frac{1}{\cos\theta}$ is undefined when $\cos\theta = 0$. Therefore, non-permissible values of $\theta$ are those where $\cos\theta = 0$, i.e., $$\theta = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}.$$ 3. **Step (b): Prove the identity for permissible $\theta$** - Start with the left-hand side (LHS): $$\frac{1 - \tan^2\theta}{\sec^2\theta}.$$ - Recall the identities: $$\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \sec\theta = \frac{1}{\cos\theta}.$$ - Substitute these into the LHS: $$\frac{1 - \left(\frac{\sin\theta}{\cos\theta}\right)^2}{\left(\frac{1}{\cos\theta}\right)^2} = \frac{1 - \frac{\sin^2\theta}{\cos^2\theta}}{\frac{1}{\cos^2\theta}}.$$ - Simplify numerator: $$1 - \frac{\sin^2\theta}{\cos^2\theta} = \frac{\cos^2\theta - \sin^2\theta}{\cos^2\theta}.$$ - So LHS becomes: $$\frac{\frac{\cos^2\theta - \sin^2\theta}{\cos^2\theta}}{\frac{1}{\cos^2\theta}}.$$ - Dividing by a fraction is multiplying by its reciprocal: $$= \frac{\cos^2\theta - \sin^2\theta}{\cos^2\theta} \times \cancel{\frac{\cos^2\theta}{1}} = \cos^2\theta - \sin^2\theta.$$ - The $\cos^2\theta$ terms cancel as shown: $$\frac{\cancel{\cos^2\theta} - \sin^2\theta}{\cancel{\cos^2\theta}} \times \frac{\cancel{\cos^2\theta}}{1} = \cos^2\theta - \sin^2\theta.$$ - Recall the double-angle identity for cosine: $$\cos 2\theta = \cos^2\theta - \sin^2\theta.$$ - Therefore, $$\text{LHS} = \cos^2\theta - \sin^2\theta = \cos 2\theta = \text{RHS}.$$ 4. **Conclusion:** - The identity holds for all $\theta$ where $\cos\theta \neq 0$. - Non-permissible values are $\theta = \frac{\pi}{2} + k\pi$, $k \in \mathbb{Z}$. **Final answer:** The identity is proven for all permissible $\theta$ and non-permissible values are where $\cos\theta=0$.