sin = sqrt(1 - cos^2)tan = sqrt(1 - cos^2)/cos
sec = 1/cos
cosec = 1/sqrt(1 - cos^2)
cot = cos/sqrt(1 - cos^2)
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There are a few ways. First, there are a multitude of trigonometric tables which list the sines and cosines of a variety of values. if you now one trigonometric value of a number, you can find all the others by hand, and you can also use a Taylor series approximation to find a fairly accurate value. (In fact, many calculators use Taylor series to find trigonometric values.)
Just as with any other identity, a trigonometric identity is a trigonometric statement (other than a definition), which is true for all values of the variable or variables.
If you want to simplify that, it usually helps to express all the trigonometric functions in terms of sines and cosines.
Yes.
yes.