These formulas express sine, cosine, tangent, and cotangent functions in terms of the half-angle tangent substitution \( t = \tan \frac{x}{2} \). This transformation simplifies trigonometric expressions, especially in integration and solving equations.
\[ \sin x = \frac{2t}{1 + t^2} \]
Explanation: Sine of \(x\) is written as a rational function of \(t\), simplifying expressions involving sine.
\[ \cos x = \frac{1 - t^2}{1 + t^2} \]
Explanation: Cosine of \(x\) is expressed in terms of \(t\), converting trigonometric functions into algebraic forms.
\[ \tan x = \frac{2t}{1 - t^2} \]
Explanation: Tangent is transformed to a rational function involving \(t\), helpful for simplifying and integrating.
\[ \cot x = \frac{1 - t^2}{2t} \]
Explanation: Cotangent is also expressed as a rational function of \(t\), facilitating algebraic manipulation.