Respuesta :

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ANSWER

[tex]\frac{( \csc x + 1)( \csc(x) - 1)}{ \cot ^{2} (x) }=1 [/tex]

EXPLANATION

The given identity is:

[tex] \frac{( \csc x + 1)( \csc(x) - 1)}{ \cot ^{2} (x) } [/tex]

Recall that:

[tex](x + 1)(x - 1) = {x}^{2} - {y}^{2} [/tex]

We apply difference of two squares to the numerator to get:

[tex] \frac{\csc ^{2} x - 1}{ \cot ^{2} (x) } [/tex]

Also recall the Pythagorean Identity.

[tex]1 + \cot^{2} (x) = \csc ^{2} (x) [/tex]

This implies that,.

[tex] \cot^{2} (x) = \csc ^{2} (x) - 1[/tex]

Hence our identity becomes:

[tex]\frac{\cot ^{2} x }{ \cot ^{2} (x) } = 1[/tex]

lublana

Answer:

Step-by-step explanation:

Given expression is [tex]\frac{\left(\csc\left(x\right)+1\right)\left(\csc\left(x\right)-1\right)}{\cot^2\left(x\right)}[/tex]

Now we need to simplify that to complete the identity.

[tex]\frac{\left(\csc\left(x\right)+1\right)\left(\csc\left(x\right)-1\right)}{\cot^2\left(x\right)}[/tex]

[tex]=\frac{\csc^2\left(x\right)+1\csc\left(x\right)-1\csc\left(x\right)-1^2}{\cot^2\left(x\right)}[/tex]

[tex]=\frac{\csc^2\left(x\right)+1\csc\left(x\right)-1\csc\left(x\right)-1}{\cot^2\left(x\right)}[/tex]

[tex]=\frac{\csc^2\left(x\right)-1}{\cot^2\left(x\right)}[/tex]

Apply formula

[tex]\csc^2\left(\theta\right)=1+\cot^2\left(\theta\right)[/tex]

[tex]=\frac{\cot^2\left(x\right)}{\cot^2\left(x\right)}[/tex]

[tex]=1[/tex]

Hence required identity is

[tex]\frac{\left(\csc\left(x\right)+1\right)\left(\csc\left(x\right)-1\right)}{\cot^2\left(x\right)}=1[/tex]

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