Answer:
The relation is:
cosec^2(a) - cot^2(a) = 1.
Where:
cosec(x) = 1/sin(x)
cot(x) = 1/tan(x) = cos(x)/sin(x)
Let's prove this.
Ok, we know by the Pythagorean theorem, that:
r^2 = x^2 + y^2
And we have the trigonometric relations, for an angle "a" measured from the x-axis
cos(a) = x/r
sin(a) = y/r
tan(a) = y/x
Where r is the hypotenuse, x and y are the catheti.
Now, let's divide the Pythagorean theorem by y^2 in both sides to get:
(r^2/y^2) = (x^2 + y^2)/y^2
r^2/x^2 = x^2/y^2 + y^2/y^2
r^2/y^2 = 1 + x^2/y^2
(r/y)^2 = 1 + (x/y)^2
and remember that:
y/r = sin(a)
then:
r/y = 1/sin(a) = cosec(a)
(r/y)^2 = cosec^2(a)
and:
y/x = tan(a)
then:
x/y = 1/tan(a) = cot(a)
and:
(x/y)^2 = cot^2(a)
replacing these in the equation (r/y)^2 = 1 + (x/y)^2 we get:
cosec^2(a) = 1 + cot^2(a)
cosec^2(a) - cot^2(a) = 1
