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Note that sec ( t ) = cosh ( t ) ⇒ sec 2 ( t ) = cosh 2 ( t ) ⇒ sec 2 ( t ) − 1 = cosh 2 ( t ) − 1 ⇒ tan 2 ( t ) = sinh 2 ( t ) ⇒ tan ( t ) = ± sinh ( t ) .
By graphing the functions and zooming in on the points, I was able to tell that sometimes when sec ( t ) = cosh ( t ) , we have tanh ( t ) = sinh ( t ) . The other times sec ( t ) = cosh ( t ) (when tan ( t ) = sinh ( t ) ), we have tan ( t ) = − sinh ( t ) . Conversely, when we have tan ( t ) = sinh ( t ) , we either get sec ( t ) = cosh ( t ) or sec ( t ) = − cosh ( t ) , depending on the value of t .