True or False? Proving Identities

Question

(1 -sin t)/(cos t) + (cos t)/(1- sin t)= (2 sec t) is an identity.

True or False?

Answer 1

The first thing I did was subtract the right side from the left side and see that it equaled 0, so I knew that this was an identity,

Then, I set to proving it.

We also recall that (1 - sin t)(1 + sin t) = 1 - sin² t = cos² t, so let's just multiply the second term by (1 + sin t)/(1 + sin t) and see what happens.

(1 - sin t)/cost + cos t/(1 - sin t) (1 + sin t)/(1 + sin t) =

(1 - sin t)/cost + cos t(1+sin t)/(1 - sin² t) =

(1 - sin t)/cost + cos t(1+sin t)/cos² t =

(1 - sin t)/cost + (1+sin t)/cos t =

[(1 - sin t) + (1 + sin t)]/cos t =

2/cos t =

2 sec t

Answer 2

(1 -sin t)/(cos t) + (cos t)/(1- sin t)

(((1 -sin t)(1 -sin t)+cos^2(t))/(cos(t)[(1 -sin t)]

(1-2sin(t)+sin^2(t)+cos^2(t))/((cos(t)[(1 -sin t)]

(2-2sin(t))/((cos(t)[(1 -sin t)

(2(1-sin(t))/(cos(t)[(1 -sin t)

2/cos(t)= 2sec(t)

TRUE, (1 -sin t)/(cos t) + (cos t)/(1- sin t)= (2 sec t) is an identity.