# How do you find the equation of the tangent line to the curve #f(x)=(secx)(tanx)# at x=pi/6?

Equation of tangent is

To find tangent line, we need to find the point at which tangent is drawn and its slope and then use point slope form of equation.

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To find the equation of the tangent line to the curve f(x) = (secx)(tanx) at x = pi/6, we need to find the derivative of the function and evaluate it at x = pi/6.

First, let's find the derivative of f(x). Using the product rule, we have:

f'(x) = (secx)(secx)(tanx) + (secx)(tanx)(tanx)

Next, we substitute x = pi/6 into f'(x) to find the slope of the tangent line at x = pi/6:

f'(pi/6) = (sec(pi/6))(sec(pi/6))(tan(pi/6)) + (sec(pi/6))(tan(pi/6))(tan(pi/6))

Now, we simplify the expression:

f'(pi/6) = (2)(2)(√3/3) + (2)(√3/3)(√3/3)

Finally, we simplify further to get the slope of the tangent line:

f'(pi/6) = 4√3/3 + 2/3

Therefore, the slope of the tangent line to the curve f(x) = (secx)(tanx) at x = pi/6 is 4√3/3 + 2/3.

To find the equation of the tangent line, we use the point-slope form of a line. We know that the point (pi/6, f(pi/6)) lies on the tangent line. Substituting x = pi/6 and f(x) = (sec(pi/6))(tan(pi/6)) into the point-slope form, we get:

y - f(pi/6) = (4√3/3 + 2/3)(x - pi/6)

Simplifying further, we have:

y - f(pi/6) = (4√3/3 + 2/3)x - (4√3/3 + 2/3)(pi/6)

Therefore, the equation of the tangent line to the curve f(x) = (secx)(tanx) at x = pi/6 is:

y = (4√3/3 + 2/3)x - (4√3/3 + 2/3)(pi/6)

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