Double Angle Identities

Double Angle Identities of Main Trigonometric Functions

Function	Formula	Description
sin(2θ)	$2\sin\theta\cos\theta$	Product of sine and cosine doubled - fundamental for wave interference
cos(2θ)	$\cos^2\theta - \sin^2\theta$	Difference of squares form. Also equals 2cos²θ - 1 or 1 - 2sin²θ - three equivalent expressions
tan(2θ)	$\displaystyle\frac{2\tan\theta}{1 - \tan^2\theta}$	Fraction form - undefined when tan²θ = 1 (at 45°, 135°, etc.)
csc(2θ)	$\displaystyle\frac{\sec\theta\csc\theta}{2}$	Reciprocal relationship - product of original secant and cosecant halved
sec(2θ)	$\displaystyle\frac{\sec^2\theta}{2\cos^2\theta - 1}$	Complex form involving both secant and cosine - used in advanced calculus
cot(2θ)	$\displaystyle\frac{\cot^2\theta - 1}{2\cot\theta}$	Quotient form - becomes undefined when cotθ = 0 (at 90°, 270°, etc.)