Maths: How tan x + cot x Simplifies to 2/sin 2x

Trigonometric identities are among the most powerful tools in Maths AA HL, letting you rewrite complicated expressions into forms that reveal their true behaviour. A single function can look quite different depending on how it is written, and recognising these equivalent forms is the key skill this topic develops. Take f(x) = tan x + cot x. Writing tan x as sin x / cos x and cot x as cos x / sin x, then combining over a common denominator, produces (sin²x + cos²x) / (sin x cos x). Since sin²x + cos²x = 1, this collapses neatly to 1 / (sin x cos x), and applying the double-angle identity sin 2x = 2 sin x cos x gives 2 / sin 2x. That transformation is what unlocks everything else: solving equations, describing endpoint behaviour, and pinning down the range all flow from this single simplified form.