If \( 0^{\circ} \leq \theta<90^{\circ} \), then \( \frac{1}{1-\sin \theta}= \) A. \( \frac{1}{\cos ^{2} \theta}+\frac{\tan \theta}{\cos \theta} \) B. \( 1-\frac{1}{\sin \theta} \) C. \( 1+\sin \theta \) D. \( \tan ^{2} \theta+\frac{\tan \theta}{\cos \theta} \) E. \( \frac{1}{\cos ^{2} \theta}+\sin ^{2} \theta \)

To solve the equation \( \frac{1}{1-\sin \theta} \), we will analyze each of the options provided to see if any of them are equivalent to this expression. ### Option A: \( \frac{1}{\cos^2 \theta} + \frac{\tan \theta}{\cos \theta} \) First, we simplify \( \frac{\tan \theta}{\cos \theta} \): \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \implies \frac{\tan \theta}{\cos \theta} = \frac{\sin \theta}{\cos^2 \theta} \] Thus, we have: \[ \frac{1}{\cos^2 \theta} + \frac{\tan \theta}{\cos \theta} = \frac{1}{\cos^2 \theta} + \frac{\sin \theta}{\cos^2 \theta} = \frac{1 + \sin \theta}{\cos^2 \theta} \] Now, we need to check if this is equal to \( \frac{1}{1 - \sin \theta} \): \[ \frac{1 + \sin \theta}{\cos^2 \theta} \text{ and } \frac{1}{1 - \sin \theta} \] Cross-multiplying gives: \[ (1 + \sin \theta)(1 - \sin \theta) = \cos^2 \theta \] Expanding the left side: \[ 1 - \sin^2 \theta = \cos^2 \theta \] This is true by the Pythagorean identity. Therefore, option A is correct. ### Option B: \( 1 - \frac{1}{\sin \theta} \) Simplifying this gives: \[ 1 - \frac{1}{\sin \theta} = \frac{\sin \theta - 1}{\sin \theta} \] This does not match \( \frac{1}{1 - \sin \theta} \). ### Option C: \( 1 + \sin \theta \) This is simply \( 1 + \sin \theta \) and does not match \( \frac{1}{1 - \sin \theta} \). ### Option D: \( \tan^2 \theta + \frac{\tan \theta}{\cos \theta} \) We know: \[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \quad \text{and} \quad \frac{\tan \theta}{\cos \theta} = \frac{\sin \theta}{\cos^2 \theta} \] Thus: \[ \tan^2 \theta + \frac{\tan \theta}{\cos \theta} = \frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\sin \theta}{\cos^2 \theta} = \frac{\sin^2 \theta + \sin \theta}{\cos^2 \theta} \] This does not match \( \frac{1}{1 - \sin \theta} \). ### Option E: \( \frac{1}{\cos^2 \theta} + \sin^2 \theta \) This simplifies to: \[ \frac{1 + \sin^2 \theta \cos^2 \theta}{\cos^2 \theta} \] This does not match \( \frac{1}{1 - \sin \theta} \). ### Conclusion The only option that matches \( \frac{1}{1 - \sin \theta} \) is: \[ \boxed{A} \]