Question

What is the equation of the tangent line of `f(x) =arcsec(tanx)` at `x=pi/4`?

Answer 1

`x = pi/4` is a bracing tangent, at turning points
`(pi/4, 2kpi), k = 0, +-1, +-2, +-3, ...` .. See illustrative graphs and explanation.

Explanation:

`y = arcsec ( tan x ) in [ 0. pi ]`

`tan x = sec y notin ( -1, 1 ),` and so,

`x notin ( -pi/4, pi/4 ),`

See illustrative extended graph using inversion for piecewise

wholesome #y = (sec)^(-1)(tan x ):
graph{cos y sin x-cos x=0}

Graph marked for range #y in [ 0, pi ] and tangent at (pi/4, 0):

Slide the graph to see how the tangent braces the extended graph.

graph{(cos y sin x-cos x)(x-0.7854+0.001y)(y-3.14 +0y)(y+0y)=0[0 6.28 0 3.14]}