Verify that each equation is an identity.
cos x = (1 - tan² (x/2))/(1 + tan² (x/2))

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Recall the half-angle identity for tangent: \(\tan\left(\frac{x}{2}\right) = t\). We want to express \(\cos x\) in terms of \(t\).

Use the double-angle identity for cosine in terms of tangent of half-angle: \(\cos x = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)}\).

Substitute \(t = \tan\left(\frac{x}{2}\right)\) into the right-hand side to get \(\frac{1 - t^2}{1 + t^2}\).

Recognize that this expression matches the given right-hand side of the equation, so the identity holds if both sides are equal for all \(x\) where defined.

To verify, you can start from the left-hand side \(\cos x\) and rewrite it using the half-angle tangent substitution, or start from the right-hand side and simplify to \(\cos x\) using Pythagorean identities.

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