Answer:
We want to prove the relation:
cosec(a)^2 - cot(a)^2 = 1
where:
cosec(a) = 1/sin(a)
cot(a) = 1/tg(a) = cos(a)/sin(a)
We can start with the relationship:
cos(a)^2 + sin(a)^2 = 1
Now, let's divide by sin(a)^2 in both sides:
(cos(a)^2 + sin(a)^2)/sin(a)^2 = 1/sin(a)^2
cos(a)^2/sin(a)^2 + sin(a)^2/sin(a)^2 = (1/sin(a))^2
(cos(a)/sin(a))^2 + 1 = (1/sin(a))^2
and remember that:
cosec(a) = 1/sin(a)
cot(a) = 1/tg(a) = cos(a)/sin(a)
Then we can write:
(cos(a)/sin(a))^2 + 1 = (1/sin(a))^2
as:
cot(a)^2 + 1 = cosec(a)^2
1 = cosec(a)^2 - cot(a)^2
And this is the relation we wanted to prove.
