How do you find the derivative of $\frac{1 - sec x}{tan x}$ $(1-secx)/tanx$ ?

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Answer 1 · 2015-05-15T13:58:24

We could use the quotient rule and differentiate the function from this form, but it may be simpler to rewrite the function first:

$f (x) = \frac{1 - sec x}{tan x} = \frac{1}{tan x} - \frac{sec x}{tan x}$ $f(x)=(1-secx)/tanx = 1/tanx -secx/tanx$

$ssss$ $color(white)"ssss"$ $= cot x - sec x cot x$ $= cotx -secxcotx$

$ssss$ $color(white)"ssss"$ $= cot x - csc x$ $= cotx - cscx$

Now it is a matter on knowing these derivatives (memorize them):

$f'(x)=-csc^2x-(-cscxcotx)=cscxcotx-csc^2x$

Or, if you prefer:

$f'(x) = cscx(cotx-cscx)$

How do you find the derivative of (1-secx)/tanx1−secxtanx(1-secx)/tanx?