How do you find the derivative of $\frac{tan x}{sin x}$ $tanx/sinx$ ?

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Answer 1 · 2016-07-15T06:17:24

$\frac{d}{d x} [\frac{tan x}{sin x}] = \frac{d}{d x} [sec x] = sec x tan x$ $d/dx[tan x/sinx] = d/dx[sec x] = sec x tan x$

Before we try to take a derivative, we could still simplify this expression.

Since $tan x = \frac{sin x}{cos x}$ $tan x = (sinx)/(cosx)$ , we can write

$tanx/(sinx) = (cancel(sinx)/cosx * 1/(cancel(sinx)))/cancel((sinx/1 * 1/sinx)) = 1/cosx = sec x$

Also, we know by definition that $d/dx[sec x] = secx tanx$ , thus giving us

$d/dx[tan x/sinx] = d/dx[sec x] = sec x tan x$

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