1.

If y= `y=sqrt((1-sin 2x )/(1+sin 2 x))` prove that `(dy)/(dx)+sec^2(pi/4-x)=0.`

Answer» `y= sqrt((1-sin 2x)/(1+sin 2x))`
`=sqrt((cos^2 x+sin^2x-2sin x cos x)/(cos^2+sin^2x+2 sin x cos x ))`
`=sqrt(((cos x-sin x)^2)/(cos x +sin x )^2)`
`=(cos x - sin x)/(cosx+sinx)`
`=(1-tanx)/(1+tan x)=(tan"(pi)/(4)tanx)/(1+tanpi/2.tanx)=tan (pi/(4)=x)`
`rArr dy/dx=d/dxtan ((pi)/4-x)`
`=sec^2((pi)/(4)-x)d/dx(pi/4-x)`
`=-sec^2(pi/4-x)`
`rArr (dy)/(dx)+sec^2(pi/4-x)=0.`


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