Expressions for sine and cosine from inverse tangent for JEE

Formula

\(\sin(\tan^{-1}x)=\dfrac{x}{\sqrt{1+x^2}},\quad \cos(\tan^{-1}x)=\dfrac{1}{\sqrt{1+x^2}}\)

Variables: \(x\in\mathbb{R}\).

Conditions: Cosine positive because \(\tan^{-1}x\in(-\pi/2,\pi/2)\).

Simplification and conversion to algebraic form.