Trigonometry Identities

The definitions of the three main trig functions; sine, cosine, and tangent:

`{:(sinx, ≡, (o p p) / (h y p)), (cosx, ≡, (a d j) / (h y p)), (tanx, ≡, sinx / cosx ≡ (o p p) / (a d j)):}`

The definitions of the inverse identities; cosecant (often also abbreviated as csc), secant, and cotangent:

`{:(\text(cosec) x, ≡, 1/sinx), ( secx, ≡, 1/cosx), ( cotx, ≡, 1/tanx):}`

Even/odd identities; sine and tangent are odd functions, cosine is an even function:

`{:(sin(-x), ≡, -sinx), (cos(-x), ≡, cosx), (tan(-x), ≡, -tanx):}`

`{:(\text(cosec)(-x), ≡, -\text(cosec) x), ( sec(-x), ≡, secx), ( cot(-x), ≡, -cotx):}`

Pythagorean identities; the main identity, then that divided by sine² and cosine² respectively:

`sin^2x + cos^2x ≡ 1`

`{:(1 + cot^2x, ≡, \text{cosec}^2 x), (1 + tan^2x, ≡, sec^2x):}`

Not a trig identity but good practise for rationalising the denominator (it's best form to have a rational number on the bottom of a quotient):

`1 / (sqrt2 + 1) + 1 ≡ sqrt2`

Trig identities practise:

`sec^4x - sec^2x ≡ tan^4x + tan^2x`