Question

If `y = (1+cos 2x)/(1-cos 2x)`how can I prove that`( 1-cos 2X)^2` multiplied by firs derivatives plus `4 sin 2x` equal zero ?

Answer 1

Given

`y = (1+cos 2x)/(1-cos 2x)`

`+>y = (2cos ^2x)/(2sin^2 x)`

`=>y = cot ^2x`

`=>(dy)/(dx) = -2cot x*csc^2x`

Now
`( 1-cos 2x)^2 xx(dy)/(dx) +4sin2x`

`=>( 2sin ^2 x)^2 xx( -2cot x*csc^2x) +4sin2x`

`=>( 2sin ^2 x)^2 xx( -2cos x/sinx*1/sin^2x) +4sin2x`

`=> 4sin ^4 x xx( -2cos x/sinx*1/sin^2x) +4sin2x`

`=> -8sin xcos +8sinxcosx=0`

Proved