Respuesta :

Stephen46

Step-by-step explanation:

First factor out the negative sign from the expression and reorder the terms

That's

[tex] \frac{1}{ - (( \tan(2A) - \tan(6A) )} - \frac{1}{ \cot(6A) - \cot(2A) } [/tex]

Using trigonometric identities

That's

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

Rewrite the expression

That's

[tex]\frac{1}{ - (( \tan(2A) - \tan(6A) )} - \frac{1}{ \frac{1}{ \tan(6A) } } - \frac{1}{ \frac{1}{ \tan(2A) } } [/tex]

We have

[tex] - \frac{1}{ \tan(2A) - \tan(6A) } - \frac{1}{ \frac{ \tan(2A) - \tan(6A) }{ \tan(6A) \tan(2A) } } [/tex]

Rewrite the second fraction

That's

[tex] - \frac{1}{ \tan(2A) - \tan(6A) } - \frac{ \tan(6A) \tan(2A) }{ \tan(2A) - \tan(6A) } [/tex]

Since they have the same denominator we can write the fraction as

[tex] - \frac{1 + \tan(6A) \tan(2A) }{ \tan(2A) - \tan(6A) } [/tex]

Using the identity

[tex] \frac{x}{y} = \frac{1}{ \frac{y}{x} } [/tex]

Rewrite the expression

We have

[tex] - \frac{1}{ \frac{ \tan(2A) - \tan(6A) }{1 + \tan(6A) \tan(2A) } } [/tex]

Using the trigonometric identity

[tex] \frac{ \tan(x) - \tan(y) }{1 + \tan(x) \tan(y) } = \tan(x - y) [/tex]

Rewrite the expression

That's

[tex] - \frac{1}{ \tan(2A -6A) } [/tex]

Which is

[tex] - \frac{1}{ \tan( - 4A) } [/tex]

Using the trigonometric identity

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

Rewrite the expression

That's

[tex] - \cot( - 4A) [/tex]

Simplify the expression using symmetry of trigonometric functions

That's

[tex] - ( - \cot(4A) )[/tex]

Remove the parenthesis

We have the final answer as

[tex] \cot(4A) [/tex]

As proven

Hope this helps you

tramserran

Answer:   see proof below

Step-by-step explanation:

Use the following identities:

[tex]\cot\alpha=\dfrac{1}{\tan\alpha}\\\\\\\cot(\alpha-\beta)=\dfrac{1+\tan\alpha\cdot \tan\beta}{\tan\alpha-\tan\beta}[/tex]

Proof  LHS →  RHS

Given:                  [tex]\dfrac{1}{\tan 6A-\tan 2A}-\dfrac{1}{\cot 6A-\cot 2A}[/tex]

Cot Identity:        [tex]\dfrac{1}{\tan 6A-\tan 2A}-\dfrac{1}{\dfrac{1}{\tan 6A}-\dfrac{1}{\tan 2A}}[/tex]

Simplify:              [tex]\dfrac{1}{\tan 6A-\tan 2A}-\dfrac{1}{\dfrac{1}{\tan 6A}\bigg(\dfrac{\tan 2A}{\tan 2A}\bigg)-\dfrac{1}{\tan 2A}\bigg({\dfrac{\tan 6A}{\tan 6A}\bigg)}}[/tex]

                         [tex]= \dfrac{1}{\tan 6A-\tan 2A}-\dfrac{1}{\dfrac{\tan 2A-\tan 6A}{\tan 6A\cdot \tan 2A}}[/tex]

                         [tex]= \dfrac{1}{\tan 6A-\tan 2A}-\dfrac{\tan6A\cdot \tan 2A}{\tan 2A-\tan 6A}[/tex]

                         [tex]= \dfrac{1}{\tan 6A-\tan 2A}-\dfrac{\tan6A\cdot \tan 2A}{\tan 2A-\tan 6A}\bigg(\dfrac{-1}{-1}\bigg)[/tex]

                        [tex]= \dfrac{1}{\tan 6A-\tan 2A}+\dfrac{\tan6A\cdot \tan 2A}{\tan 6A-\tan 2A}[/tex]

                        [tex]= \dfrac{1+\tan6A\cdot \tan 2A}{\tan 6A-\tan 2A}[/tex]

Sum Difference Identity:    cot(6A - 2A)

Simplify:                               cot 4A

cot 4A = cot 4A   [tex]\checkmark[/tex]

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