 #### Where do the trigonometric identities comes from?

By definition the side opposite an angle in a right-angle triangle, with unit length hypotenuse, is the sine of θ, written sinθ. Similarly, the side opposite the complimentary angle of θ, coθ=90–θ, would be sin(coθ). It soon became preferred to call the side adjacent to the angle, the cosine of θ, written cosθ. Hence cosθ=sin(coθ)=sin(90–θ).

Let us consider this in the context of a unit circle. By definition, the length of the adjacent side will be cosθ and the length of the opposite side will be sinθ. And so we have a more general definition of sine and cosine: For any point, P, on a unit circle defined by an angle θ, in an anticlockwise direction from the x axis, the co-ordinate of P is (cosθ, sinθ). The advantage of this definition is that it allows us to calculate sines and cosines of angles greater than 90o and provides a meaningful context for them.

As this definition only allows us to directly express a relationship between the hypotenuse and either the opposite or the adjacent side, it was decided that one further trigonometric definition would be useful, which linked the opposite and adjacent sides directly. A line that cuts a circle at two points is called a secant and a line that cuts a circle at precisely one point is called a tangent. So for any right angle triangle with a base length 1 unit, the length of the side opposite the angle is defined as tanθ and the hypotenuse is secθ.

One further definition is added for cotangent (cot) and cosecant (cosec). It is hardly surprising that the six trigonometric ratios are related. Using similar triangles with the first and second triangles, sinθ/cosθ=tanθ/1; that is tanθ=sinθ/cosθ.

In the same we we can show that secθ=1/cosθ, cosecθ=1/sinθ, and cotθ=1/tanθ.

Using the Pythagorean Theorem in each triangle we get three identities:

sin2θ+cos2θ=1
tan2θ+1=sec2θ
cot2θ+1=cosec2θ