Question

Calculate the equation of the tangent and normal lines for the curve f(x) = tan 2x, at the point where the x-coordinate is: x = π/8.

Answer 1

dy/dx = 2sec^2 (2x)
when x = π/8
f(π/8) = tan π/4 = 1
so our point of contact is (π/8, 1)

slope of tangent = 2 sec^2 (π/4)
= 2(√2/1)^2 = 4

equation of tangent:
y - 1 = 4(x - π/8)
y - 1 = 4x - π/2
y = 4x + 1-π/2

normal has slope -1/4
y = (-1/4)x + b
but the point (π/8,1) lies on it, so
1 = (-1/4)(π/8) + b
b = 1 + π/32
y = (-1/4)x + 1+π/32