trig derivative

Question

What is the derivative for y=tan⁡(x) is dy/dx=sec^2⁡x. Verify this using the identity tan⁡x=sin⁡x/cos⁡x .

show all steps please, or anything that may help me understand better.

Answer 1

We can use the identity tanx = sinx/cosx and then differentiate using quotient rule (and the fact that d/dx[sinx] = cosx and d/dx[cosx] = -sinx):

y = sinx / cosx

dy/dx = [cosx·cosx - (-sinx)(sinx)] / cos²x [(deriv of top)·bottom - (deriv of bottom)·top] / bottom²

= [cos²x + sin²x] / cos²x

= 1 / cos²x

= sec²x (since secx = 1/cosx)