How do you simplify the expression $sec x cot x - cot x cos x$ $secxcotx-cotxcosx$ ?

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How do you simplify the expression $sec x cot x - cot x cos x$ $secxcotx-cotxcosx$ ?

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Answer 1 · 2016-08-28T02:00:33

$sec x cot x - cot x cos x = sin x$ $secxcotx - cotxcosx=sinx$ for all $x$ $x$ such that $cos x, sin x \neq 0$ $cosx, sinx != 0$

Explanation:

We will use the following:

$sec x = \frac{1}{cos x}$ $secx = 1/cosx$
$cot x = \frac{cos x}{sin x}$ $cotx = cosx/sinx$
${sin}^{2} x + {cos}^{2} x = 1 \Rightarrow 1 - {cos}^{2} x = {sin}^{2} x$ $sin^2x + cos^2x = 1 => 1 - cos^2x = sin^2x$

$sec x cot x - cot x cos x = \frac{1}{cos x} \cdot \frac{cos x}{sin x} - \frac{cos x}{sin x} \cdot cos x$ $secxcotx - cotxcosx = 1/cosx*cosx/sinx - cosx/sinx*cosx$

$= \frac{1}{sin x} - \frac{{cos}^{2} x}{sin x}$ $=1/sinx-cos^2x/sinx$ (assuming $cos x \neq 0$ $cosx != 0$ )

$= \frac{1 - {cos}^{2} x}{sin x}$ $=(1-cos^2x)/sinx$

$= \frac{{sin}^{2} x}{sin x}$ $=sin^2x/sinx$

$= sin x$ $=sinx$ (assuming $sin x \neq 0$ $sinx != 0$ )

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How do you simplify the expression $sec x cot x - cot x cos x$ $secxcotx-cotxcosx$ ?

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