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`(cos4A + cos2A)/(sin4A - sin2A)` =`(1)/(tan A)`  Prove the double angle identity

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rockstar195699 | Student, Undergraduate | eNoter

Posted April 30, 2013 at 3:04 PM via web

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`(cos4A + cos2A)/(sin4A - sin2A)` =`(1)/(tan A)` 

Prove the double angle identity

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted April 30, 2013 at 3:13 PM (Answer #1)

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The identity `(cos 4A + cos 2A)/(sin 4A - sin 2A) = 1/tan A` has to be proved.

`(cos 4A + cos 2A)/(sin 4A - sin 2A)`

= `(2*cos^2 2A + cos 2A - 1)/(2*sin 2A*cos 2A - sin 2A)`

= `(2*cos^2 2A + 2cos 2A - cos 2A - 1)/(2*sin 2A*cos 2A - sin 2A)`

= `(2*cos 2A(cos 2A + 1) - 1(cos 2A + 1))/(2*sin 2A*cos 2A - sin 2A)`

= `((2*cos 2A-1)(cos 2A + 1))/(sin 2A*(2*cos 2A - 1))`

= `(cos 2A + 1)/(sin 2A)`

= `(2*cos^2A -1 + 1)/(2*sin A*cos A)`

= `(2*cos^2A)/(2*sin A*cos A)`

= `(cos A)/(sin A)`

= `1/tan A`

This proves that `(cos 4A + cos 2A)/(sin 4A - sin 2A) = 1/tan A`

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