Question

Cos sec² a - cot²a=1 ​

Answer 1

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.