Evaluate: lim_x→π/2 (secx - tanx)

Put y = π/2 - x. Then y→0 as x → π/2

lim_x→π/2 (secx - tanx) = lim_y→0 [ sec(π/2 - y) - tan(π/2 - y)]

= lim_y→0 (cscy - coty)

= lim_y→0 [(1/sin y) - (cos y/sin y)]

= lim_y→0 [(1 - cos y)/sin y]

= lim_y→0 [(2 sin²(y/2)) / (2 sin(y/2) cos(y/2))]

= lim_y→0 tan(y/2) = 0