Proving trigonomeric identities \( \frac{1}{1-\sin x}-\frac{1}{1+\sin x}=\frac{2 \tan x}{\cos x} \)

To transform the left side into the right side, start by finding a common denominator for the fractions. The common denominator for \(1 - \sin x\) and \(1 + \sin x\) is \((1 - \sin x)(1 + \sin x)\). Combine the fractions: \[ \frac{(1 + \sin x) - (1 - \sin x)}{(1 - \sin x)(1 + \sin x)} = \frac{2\sin x}{1 - \sin^2 x} \] Utilizing the Pythagorean identity that \(1 - \sin^2 x = \cos^2 x\), we can rewrite the fraction: \[ = \frac{2\sin x}{\cos^2 x} \] Since \( \tan x = \frac{\sin x}{\cos x} \), this expression becomes: \[ = \frac{2 \tan x}{\cos x} \] And voilà! We have successfully proven the identity. One common mistake in proving trigonometric identities is neglecting to simplify expressions or misapplying Pythagorean identities. Always double-check your steps to ensure each transformation is valid. If you find yourself stuck, try rewriting in terms of sine and cosine or factor expressions to simplify the problem! If you are intrigued by trigonometric identities, there are plenty of resources online that delve deeper into advanced trigonometric techniques and their applications. Websites like Khan Academy or math-centric blogs often provide rich insights, while textbooks can offer structured learning paths with exercises to hone your skills. Don't be shy; diving deeper will only make you a trigonometric aficionado!