Verify that each equation is an identity.
cot² (x/2) = (1 + cos x)²/(sin² x)

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Recall the half-angle identity for cotangent: \(\cot\left(\frac{x}{2}\right) = \frac{1 + \cos x}{\sin x}\). This is a key formula to start with.

Square both sides of the half-angle identity to express \(\cot^2\left(\frac{x}{2}\right)\): \(\cot^2\left(\frac{x}{2}\right) = \left(\frac{1 + \cos x}{\sin x}\right)^2\).

Rewrite the right-hand side explicitly as \(\frac{(1 + \cos x)^2}{\sin^2 x}\) to match the given expression.

Since both sides are now expressed as \(\cot^2\left(\frac{x}{2}\right)\) and \(\frac{(1 + \cos x)^2}{\sin^2 x}\), conclude that the equation holds true for all \(x\) where the expressions are defined.

Optionally, verify the domain restrictions where \(\sin x \neq 0\) to ensure the identity is valid and no division by zero occurs.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing both sides simplify to the same expression, often by using fundamental identities like Pythagorean or angle formulas.

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Half-angle formulas express trigonometric functions of half an angle in terms of the full angle. For example, cot(x/2) can be rewritten using cosine and sine of x, which helps in transforming and simplifying expressions involving half angles.

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Simplifying or verifying identities often requires algebraic skills such as factoring, expanding, and rewriting expressions. Combining these with trigonometric formulas allows one to transform one side of the equation to match the other.

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