# Differentiate tan^(-1)(sqrt(√(1-x^2)/x) with respect to cos^(-1)(2x√(1-x^2)) ,when x!=0 - Mathematics

Differentiate tan^(-1)(sqrt(1-x^2)/x) with respect to cos^(-1)(2xsqrt(1-x^2)) ,when x!=0

#### Solution

Let

Putting x=cosθ, we get:

y =tan^(-1)(sqrt(1-cos^2theta)/costheta)=tan^(-1)(sintheta/costheta)=tan^(-1)(tantheta)=theta

y = cos^(−1)x

On differentiating with respect to x, we get:

dy/dx=-1/sqrt(1-x^2).........(1)

Now assume that

z=cos^(-1)(2xsqrt(1-x^2))

z=cos^(-1)(2sintheta costheta)=cos^(-1)(sin2theta)=cos^(-1)(cos(pi/2-2theta))=pi/2-2theta=pi/2-2cos^(-1)x

On differentiating with respect to x, we get:

dz/dx=2/sqrt(1-x^2)............(2)

We know that,

dy/dz=(dy/dx)/(dz/dx)

So, from equations (1) and (2), we get:

dy/dz=(-1/sqrt(1-x^2))/(2/sqrt(1-x^2))=-1/2

Derivative of tan^(-1)(sqrt(1-x^2)/x) with respect to cos^(-1)(2xsqrt(1-x^2))  is -1/2

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