# Proof of the Identity: \(\frac{1 + \tan^2 x}{2 \tan x} = \csc(2x)\) ## Step 1: Recognize a Pythagorean Identity **Statement:** Use the identity \(1 + \tan^2 x = \sec^2 x\) **Rule:**ythagorean \[ \frac{1 + \tan^2 x}{2 \tan x} = \frac{\^2 x}{2 \tan x} \] --- ## Step 2: Rewrite \(\sec^2 x\) and \(\tan x\) **Statement:** Express \(\sec x\) and \(\tan x\) in terms of sine and cosine **Rule:** Definitions of secant and tangent \[ \frac{\sec^2 x}{2 \tan x} = \frac{\left(\frac{1}{\cos x}\right)^2}{2 \left(\frac{\sin x}{\cos x}\right)} = \frac{\frac{1}{\cos^2 x}}{2 \frac{\sin x}{\cos x}} \] --- ## Step 3: Simplify the complex fraction **Statement:** Invert and multiply to simplify the fraction **Rule:** Division of fractions \[ = \frac{1}{\cos^2 x} \times \frac{\cos x}{2 \sin x} = \frac{1}{\cos^2 x} \times \frac{\cos x}{2 \sin x} \] \[ = \frac{\cos x}{2 \sin x \cos^2 x} \] --- ## Step 4: Simplify numerator and denominator **Statement:** Cancel common factors **Rule:** Simplification of fractions \[ = \frac{\cos x}{2 \sin x \cos^2 x} = \frac{1}{2 \sin x \cos x} \] --- ## Step 5: Recognize double angle identity for sine **Statement:** Recall that \(\sin 2x = 2 \sin x \cos x\) **Rule:** Double angle identity for sine \[ \frac{1}{2 \sin x \cos x} = \frac{1}{\sin 2x} \] --- ## Step 6: Express as cosecant **Statement:** Use the reciprocal relationship \(\csc \theta = \frac{1}{\sin \theta}\) **Rule:** Definition of cosecant \[ \frac{1}{\sin 2x} = \csc 2x \] --- # **Conclusion** \[ \boxed{\frac{1 + \tan^2 x}{2 \tan x} = \csc 2x} \] **The identity is proved.**