How do you simplify ${(\frac{cot x}{csc x})}^{2} + \frac{1}{{csc}^{2} x}$ $(cotx/cscx)^2+1/csc^2x$ ?

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How do you simplify ${(\frac{cot x}{csc x})}^{2} + \frac{1}{{csc}^{2} x}$ $(cotx/cscx)^2+1/csc^2x$ ?

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Answer 1 · 2016-10-12T00:11:37

${(\frac{cot x}{csc x})}^{2} + \frac{1}{{csc}^{2} x} \equiv 1$ $(cotx/cscx)^2+1/csc^2x -= 1$

Explanation:

$cot x = \frac{1}{tan x}$ $cotx=1/tanx$ ; $csc x = \frac{1}{sin x}$ $cscx=1/sinx$ ; so:

${(\frac{cot x}{csc x})}^{2} + \frac{1}{{csc}^{2} x} = {(\frac{\frac{1}{tan x}}{\frac{1}{sin x}})}^{2} + \frac{1}{\frac{1}{{sin}^{2} x}}$ $(cotx/cscx)^2+1/csc^2x = ((1/tanx)/(1/sinx))^2 + 1/(1/sin^2x)$
$= {(\frac{sin x}{tan x})}^{2} + {sin}^{2} x$ $= (sinx/tanx)^2+sin^2x$
$= {sin}^{2} x (\frac{1}{{tan}^{2} x} + 1)$ $= sin^2x (1/tan^2x+1)$
$= {sin}^{2} x ({cot}^{2} x + 1)$ $= sin^2x (cot^2x+1)$
$= {sin}^{2} x ({csc}^{2} x)$ $= sin^2x (csc^2x)$
$= {sin}^{2} x (\frac{1}{{sin}^{2} x})$ $= sin^2x (1/sin^2x)$
$= 1$ $= 1$

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How do you simplify ${(\frac{cot x}{csc x})}^{2} + \frac{1}{{csc}^{2} x}$ $(cotx/cscx)^2+1/csc^2x$ ?

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Explanation:

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How do you simplify ${(\frac{cot x}{csc x})}^{2} + \frac{1}{{csc}^{2} x}$ $(cotx/cscx)^2+1/csc^2x$ ?