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Please let me know if I'm missing something but, contrary to what's been said above, it doesn't appear that the following two expressions are equal:

e1 = -Log[Cos[t/2] - Sin[t/2]] + Log[Cos[t/2] + Sin[t/2]]

e2 = Log[Sec[t] + Tan[t]]

e1 /. t -> Pi

gives: -i Pi

e2 /. t -> Pi

gives: i Pi

e1 /. t -> Pi/2.

gives: 37.0834

e2 /. t -> Pi/2.

gives: 38.025

EDIT: I think I partly figured it out -- when t=Pi, the arguments of the logs (after using the identity log(x) - log(y) = log(x/y) are, in both cases, -1, so we get -i Pi in one case, and i Pi in the other (because of the negative sign):

(Cos[t/2] + Sin[t/2])/(Cos[t/2] - Sin[t/2]) /. t -> Pi

gives -1

Sec[t] + Tan[t] /. t -> Pi

gives -1

And when t=Pi/2, the arguments of the logs are some very large numbers, so we may be getting round-off errors.

Please let me know if I'm missing something but, contrary to what's been said above, it doesn't appear that the following two expressions are equal:

e1 = -Log[Cos[t/2] - Sin[t/2]] + Log[Cos[t/2] + Sin[t/2]]

e2 = Log[Sec[t] + Tan[t]]

e1 /. t -> Pi

gives: -i Pi

e2 /. t -> Pi

gives: i Pi

e1 /. t -> Pi/2.

gives: 37.0834

e2 /. t -> Pi/2.

gives: 38.025

EDIT: When t=Pi/2, the arguments of the logs are some very large numbers, so we may be getting round-off errors.

Please let me know if I'm missing something but, contrary to what's been said above, it doesn't appear that the following two expressions are equal:

e1 = -Log[Cos[t/2] - Sin[t/2]] + Log[Cos[t/2] + Sin[t/2]]

e2 = Log[Sec[t] + Tan[t]]

e1 /. t -> Pi

gives: -i Pi

e2 /. t -> Pi

gives: i Pi

e1 /. t -> Pi/2.

gives: 37.0834

e2 /. t -> Pi/2.

gives: 38.025

EDIT: I think I partly figured it out -- when t=Pi, the arguments of the logs are, in both cases, -1, so we get -i Pi in one case, and i Pi in the other (because of the negative sign).
And when t=Pi/2, the arguments of the logs are some very large numbers, so we may be getting round-off errors.

Please let me know if I'm missing something but, contrary to what's been said above, it doesn't appear that the following two expressions are equal:

e1 = -Log[Cos[t/2] - Sin[t/2]] + Log[Cos[t/2] + Sin[t/2]]

e2 = Log[Sec[t] + Tan[t]]

e1 /. t -> Pi

gives: -i Pi

e2 /. t -> Pi

gives: i Pi

e1 /. t -> Pi/2.

gives: 37.0834

e2 /. t -> Pi/2.

gives: 38.025

EDIT: I think I partly figured it out -- when t=Pi, the arguments of the logs (after using the identity log(x) - log(y) = log(x/y) are, in both cases, -1, so we get -i Pi in one case, and i Pi in the other (because of the negative sign):

(Cos[t/2] + Sin[t/2])/(Cos[t/2] - Sin[t/2]) /. t -> Pi

gives -1

Sec[t] + Tan[t] /. t -> Pi

gives -1

And when t=Pi/2, the arguments of the logs are some very large numbers, so we may be getting round-off errors.

Please let me know if I'm missing something but, contrary to what's been said above, it doesn't appear that the following two expressions are equal:

e1 = -Log[Cos[t/2] - Sin[t/2]] + Log[Cos[t/2] + Sin[t/2]]

e2 = Log[Sec[t] + Tan[t]]

e1 /. t -> Pi

gives: -i Pi

e2 /. t -> Pi

gives: i Pi

e1 /. t -> Pi/2.

gives: 37.0834

e2 /. t -> Pi/2.

gives: 38.025

Please let me know if I'm missing something but, contrary to what's been said above, it doesn't appear that the following two expressions are equal:

e1 = -Log[Cos[t/2] - Sin[t/2]] + Log[Cos[t/2] + Sin[t/2]]

e2 = Log[Sec[t] + Tan[t]]

e1 /. t -> Pi

gives: -i Pi

e2 /. t -> Pi

gives: i Pi

e1 /. t -> Pi/2.

gives: 37.0834

e2 /. t -> Pi/2.

gives: 38.025

EDIT: I think I partly figured it out -- when t=Pi, the arguments of the logs are, in both cases, -1, so we get -i Pi in one case, and i Pi in the other (because of the negative sign).
And when t=Pi/2, the arguments of the logs are some very large numbers, so we may be getting round-off errors.