TRIGONOMENTRY IDEANTITIES

Summary of trigonometric identities

You have seen quite a few trigonometric identities in the past few pages. It is convenient to have a summary of them for reference. These identities mostly refer to one angle denoted t, but there are a few of them involving two angles, and for those, the other angle is denoted s..
More important identities
You don't have to know all the identities off the top of your head. But these you should.

Defining relations for tangent, cotangent, secant, and cosecant in terms of sine and cosine.

tan t = sin tcos t cot t = 1tan t = cos tsin t
sec t = 1cos t csc t = 1sin t

The Pythagorean formula for sines and cosines.

sin2 t + cos2 t = 1

Identities expressing trig functions in terms of their complements

cos t = sin( pi/2 – t) sin t = cos( pi/2 – t)

cot t = tan( pi/2 – t) tan t = cot( pi/2 – t)

csc t = sec( pi/2 – t) sec t = csc( pi/2 – t)

Periodicity of trig functions. Sine, cosine, secant, and cosecant have period 2 pi while tangent and cotangent have period pi.

sin (t + 2 pi) = sin t

cos (t + 2 pi) = cos t

tan (t + pi) = tan t

Identities for negative angles. Sine, tangent, cotangent, and cosecant are odd functions while cosine and secant are even functions.

sin –t = –sin t

cos –t = cos t

tan –t = –tan t

Sum formulas for sine and cosine

sin (s + t) = sin s cos t + cos s sin t

cos (s + t) = cos s cos t – sin s sin t

Double angle formulas for sine and cosine

sin 2t = 2 sin t cos t

cos 2t = cos2 t – sin2 t = 2 cos2 t – 1 = 1 – 2 sin2 t

Less important identities
You should know that there are these identities, but they are not as important as those mentioned above. They can all be derived from those above, but sometimes it takes a bit of work to do so.

The Pythagorean formula for tangents and secants.

sec2 t = 1 + tan2 t

Identities expressing trig functions in terms of their supplements

sin( pi – t) = sin t

cos( pi – t) = –cos t

tan( pi – t) = –tan t

Difference formulas for sine and cosine

sin (s – t) = sin s cos t – cos s sin t

cos (s – t) = cos s cos t + sin s sin t

Sum, difference, and double angle formulas for tangent

tan (s + t) = tan s + tan t1 – tan s tan t
tan (s – t) = tan s – tan t1 + tan s tan t
tan 2t = 2 tan t1 – tan2 t

Half-angle formulas

sin t/2 = ± sqrt((1 – cos t) / 2)

cos t/2 = ± sqrt((1 + cos t) / 2)

tan t/2 = sin t1 + cos t = 1 – cos tsin t

Truly obscure identities
These are just here for perversity. Yes, of course, they have some applications, but they're usually narrow applications, and they could just as well be forgotten until, if ever, needed.

Product-sum identities

sin s + sin t = 2 sin s + t2 cos s – t2
sin s – sin t = 2 cos s + t2 sin s – t2
cos s + cos t = 2 cos s + t2 cos s – t2
cos s – cos t = –2 sin s + t2 sin s – t2

Product identities

sin s cos t = sin (s + t) + sin (s – t)2
cos s cos t = cos (s + t) + cos (s – t)2
sin s sin t = cos (s – t) – cos (s + t) 2
Aside: weirdly enough, these product identities were used before logarithms to perform multiplication. Here's how you could use the second one. If you want to multiply x times y, use a table to look up the angle s whose cosine is x and the angle t whose cosine is y. Look up the cosines of the sum s + t, and the difference s – t. Average those two cosines. You get the product xy! Three table look-ups, and computing a sum, a difference, and an average rather than one multiplication. Tycho Brahe (1546-1601), among others, used this algorithm known as prosthaphaeresis.

Triple angle formulas. You can easily reconstruct these from the addition and double angle formulas.

sin 3t = 3 sin t – 4 sin3 t

cos 3t = 4 cos3 t –3 cos t

tan 3t = 3 tan t – tan3t1 – 3 tan2t

More half-angle formulas. (These are used in calculus for a particular kind of substitution in integrals sometimes called the Weierstrass t-substitution.)

sin t = 2 tan t/21 + tan2 t/2
cos t = 1 – tan2 t/21 + tan2 t/2
tan t = 2 tan t/21 – tan2 t/2