How do you identify $\frac{1}{csc θ + 1} - \frac{1}{csc θ - 1}$ $1/(csctheta+1)-1/(csctheta-1)$ ?

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How do you identify $\frac{1}{csc θ + 1} - \frac{1}{csc θ - 1}$ $1/(csctheta+1)-1/(csctheta-1)$ ?

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Answer 1 · 2017-04-19T10:10:17

$- 2 {tan}^{2} θ$ $-2tan^2theta$

Explanation:

I'm going to assume this needs to be simplified:

$\frac{1}{csc θ + 1} - \frac{1}{csc θ - 1}$ $1/(csctheta+1)-1/(csctheta-1)$

$\frac{1}{\frac{1}{sin θ} + 1} - \frac{1}{\frac{1}{sin θ} - 1}$ $1/(1/sintheta+1)-1/(1/sintheta-1)$

$\frac{1}{\frac{1}{sin θ} + \frac{sin θ}{sin θ}} - \frac{1}{\frac{1}{sin θ} - \frac{sin θ}{sin θ}}$ $1/(1/sintheta+sintheta/sintheta)-1/(1/sintheta-sintheta/sintheta)$

$\frac{1}{\frac{1 + sin θ}{sin θ}} - \frac{1}{\frac{1 - sin θ}{sin θ}}$ $1/((1+sintheta)/sintheta)-1/((1-sintheta)/sintheta)$

$\frac{sin θ}{1 + sin θ} - \frac{sin θ}{1 - sin θ}$ $sintheta/(1+sintheta)-sintheta/(1-sintheta)$

$\frac{sin θ}{1 + sin θ} (\frac{1 - sin θ}{1 - sin θ}) - \frac{sin θ}{1 - sin θ} (\frac{1 + sin θ}{1 + sin θ})$ $sintheta/(1+sintheta)((1-sintheta)/(1-sintheta))-sintheta/(1-sintheta)((1+sintheta)/(1+sintheta))$

$\frac{sin θ (1 - sin θ)}{(1 + sin θ) (1 - sin θ)} - \frac{sin θ (1 + sin θ)}{(1 - sin θ) (1 + sin θ)}$ $(sintheta(1-sintheta))/((1+sintheta)(1-sintheta))-(sintheta(1+sintheta))/((1-sintheta)(1+sintheta))$

$\frac{sin θ - {sin}^{2} θ}{(1 + sin θ) (1 - sin θ)} - \frac{sin θ + {sin}^{2} θ}{(1 - sin θ) (1 + sin θ)}$ $(sintheta-sin^2theta)/((1+sintheta)(1-sintheta))-(sintheta+sin^2theta)/((1-sintheta)(1+sintheta))$

$\frac{sin θ - sin θ - {sin}^{2} θ - {sin}^{2} θ}{(1 + sin θ) (1 - sin θ)}$ $(sintheta-sintheta-sin^2theta-sin^2theta)/((1+sintheta)(1-sintheta))$

$\frac{- 2 {sin}^{2} θ}{(1 - {sin}^{2} θ)}$ $(-2sin^2theta)/((1-sin^2theta))$

Recall that ${sin}^{2} θ + {cos}^{2} θ = 1 \Rightarrow 1 - {sin}^{2} θ = {cos}^{2} θ$ $sin^2theta+cos^2theta=1=>1-sin^2theta=cos^2theta$

$\frac{- 2 {sin}^{2} θ}{{cos}^{2} θ} = - 2 {tan}^{2} θ$ $(-2sin^2theta)/cos^2theta=-2tan^2theta$

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How do you identify $\frac{1}{csc θ + 1} - \frac{1}{csc θ - 1}$ $1/(csctheta+1)-1/(csctheta-1)$ ?

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