Final answer:
The equation of the tangent line to the curve y = tan²(x) at the point (π/4, 1) is y = 4x - π + 1.
Step-by-step explanation:
To find an equation of the tangent line to the curve y = tan²(x) at the point (π/4, 1), the first step is to calculate the derivative of y with respect to x, which will give the slope of the tangent line at any point on the curve. The derivative of y = tan²(x) with respect to x is 2tan(x)sec²(x). At x = π/4, tan(π/4) = 1 and sec(π/4) = √2, so the slope (m) of the tangent line at x = π/4 is m = 2(1)(√2)² = 4.
Using the point-slope form of the line, which is y - y1 = m(x - x1), where (x1, y1) is the point of tangency and m is the slope, we can plug in the values to get the equation of the tangent line: y - 1 = 4(x - π/4). Simplifying this equation yields y = 4x - π + 1 as the equation of the tangent line to the curve at the point (π/4, 1).