Start by finding the point #(x, y)# where the function meets the tangent.
#f(pi/4) = sin(pi/4)/cos(pi/4) - cot(pi/4)#
#f(pi/4) = sin(pi/4)/cos(pi/4) - cos(pi/4)/sin(pi/4)#
#f(pi/4) = (1/sqrt(2))/(1/sqrt(2)) - (1/sqrt(2))/(1/sqrt(2))#
#f(pi/4) = 1/sqrt(2) xx sqrt(2)/1 - 1/sqrt(2) xx sqrt(2)/1#
#f(pi/4) = 1 - 1#
#f(pi/4) = 0#
We now differentiate the function.
#f(x) = sinx/cosx - cosx/sinx#
#f'(x) = (cosx xx cosx - (sinx xx -sinx))/(cosx)^2 - (-sinx xx sinx - (cosx xx cosx))/(sinx)^2#
#f'(x) = (cos^2x + sin^2x)/cos^2x - (-cos^2x - sin^2x)/sin^2x#
#f'(x) = 1/cos^2x - (-(cos^2x + sin^2x))/sin^2x #
#f'(x) = sec^2x + csc^2x#
Now, the slope of the tangent can be found by evaluating #x = a# within the derivative.
#f'(pi/4) = sec^2(pi/4) + csc^2(pi4)#
#f'(pi/4) = 1/cos^2(pi/4) + 1/sin^2(pi/4)#
#f'(pi/4) = 1/((1/sqrt(2))^2) + 1/(1/sqrt(2))^2#
#f'(pi/4) = 1/(1/2) + 1/(1/2)#
#f'(pi/4) = 2 + 2#
#f'(pi/4) = 4#
We now know the point of contact and the slope, so we can find the equation of the tangent line.
#y - y_ 1= m(x- x_1)#
#y - 0 = 4(x - pi/4)#
#y - 0 = 4x - pi#
#y = 4x - pi#
Hopefully this helps!
