Question

√1-cos2x/1+cos2x = tan x​

Answer 1

Answer:

Hope u like my process

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NOTE:-

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=> sin²x + cos²x = 1

=> cos²x - sin²x = cos2x

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\begin{lgathered}= > y = \frac{ \sqrt{1 + \cos(2x) } }{ \sqrt{1 - \cos(2x) } } \\ \\ or. \: \: y = \sqrt{ \frac{( \sin ^{2} (x) + \cos ^{2} (x) ) + ( \cos ^{2} (x) - \sin ^{2} (x) ) }{ (\sin ^{2} (x) + \cos ^{2} (x) ) - ( \cos ^{2} (x) - \sin ^{2} ( x )) } } \\ \\ or . \: \: y = \sqrt{ \frac{2 \cos ^{2} (x) }{2 \sin ^{2} (x) } } = \frac{ \cos(x) }{ \sin(x) } = \cot(x) \\ \\\end{lgathered}=>y=1−cos(2x)1+cos(2x)or.y=(sin2(x)+cos2(x))−(cos2(x)−sin2(x))(sin2(x)+cos2(x))+(cos2(x)−sin2(x))or.y=2sin2(x)2cos2(x)=sin(x)cos(x)=cot(x)

Now..

Differentiating both sides we get

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\frac{dy}{dx} = \frac{d( \cot(x) )}{dx} = - \csc ^{2} (x)dxdy=dxd(cot(x))=−csc2(x)