2. Prove the foilocing \( : \frac{1}{1+\sin \theta}+\frac{1}{1-\sin \theta}=\frac{2}{\cos ^{2} \theta} \)

1. Write the expression: \[ \frac{1}{1+\sin \theta} + \frac{1}{1-\sin \theta} \] 2. Find a common denominator. Notice that: \[ (1+\sin \theta)(1-\sin \theta) = 1 - \sin^2 \theta \] by the difference of squares formula. 3. Express both terms with the common denominator: \[ \frac{1}{1+\sin \theta} = \frac{1-\sin \theta}{(1+\sin \theta)(1-\sin \theta)} = \frac{1-\sin \theta}{1-\sin^2 \theta} \] \[ \frac{1}{1-\sin \theta} = \frac{1+\sin \theta}{(1-\sin \theta)(1+\sin \theta)} = \frac{1+\sin \theta}{1-\sin^2 \theta} \] 4. Combine the fractions: \[ \frac{1-\sin \theta}{1-\sin^2 \theta} + \frac{1+\sin \theta}{1-\sin^2 \theta} = \frac{(1-\sin \theta) + (1+\sin \theta)}{1-\sin^2 \theta} \] 5. Simplify the numerator: \[ (1-\sin \theta) + (1+\sin \theta) = 1 + 1 = 2 \] So the expression becomes: \[ \frac{2}{1-\sin^2 \theta} \] 6. Recognize that: \[ 1-\sin^2 \theta = \cos^2 \theta \] by the Pythagorean identity. 7. Substitute to obtain: \[ \frac{2}{\cos^2 \theta} \] Thus, we have proven that: \[ \frac{1}{1+\sin \theta}+\frac{1}{1-\sin \theta}=\frac{2}{\cos^2 \theta} \]