Question

If y=sqrt((1-cos4x)/(1+cos4x)) then prove that (dy)/(dx)=2sec^(2)(2x)​

Answer 1

$\underline{\textsf{Given:}}$

$\mathsf{y=\displaystyle\sqrt{\dfrac{1-cos4x}{1+cos4x}}}$

$\underline{\textsf{To prove:}}$

$\mathsf{\dfrac{dy}{dx}=2\,sec^22x}$

$\underline{\textsf{Solution:}}$

$\textsf{First we simplify the given function by using}$

$\textsf{suitable trigonometric identities}$

$\textsf{Consider,}$

$\mathsf{y=\displaystyle\sqrt{\dfrac{1-cos4x}{1+cos4x}}}$

$\mathsf{y=\displaystyle\sqrt{\dfrac{1-cos2(2x)}{1+cos2(2x)}}}$

$\textsf{Using,}$

$\boxed{\mathsf{cos2A=1-2\,sin^2A=2\,cos^2A-1}}$

$\mathsf{y=\displaystyle\sqrt{\dfrac{1-(1-2\,sin^22x)}{1+(2\,cos^22x-1)}}}$

$\mathsf{y=\displaystyle\sqrt{\dfrac{2\,sin^22x}{2\,cos^22x}}}$

$\mathsf{y=\displaystyle\sqrt{\dfrac{sin^22x}{cos^22x}}}$

$\mathsf{y=\displaystyle\sqrt{tan^22x}}$

$\mathsf{y=tan2x}$

$\textsf{Now, differentiate y with respect to x by using chain rule}$

$\mathsf{\dfrac{dy}{dx}=sec^22x\,\dfrac{d(2x)}{dx}}$

$\mathsf{\dfrac{dy}{dx}=sec^22x\,(2)}$

$\implies\boxed{\mathsf{\dfrac{dy}{dx}=2\,sec^22x}}$

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