Question

if f:R to R is defined by f(x)=1-x square/1+x square,then show that f(tan theta)=cos 2theta​

Answer 1

please see the attachment

Answer 2

The value of f(tan θ) is cos2θ.

Given,

A function f:R to R is defined by f(x) = (1-x²)/(1+x²).

To Prove,

The value of function f(tan θ) = cos2θ.

Solution,

The given function is

f(x) = (1-x²)/(1+x²)

Now, substituting the value of x as tan θ.

f(tan θ) = (1-tan²θ)/(1+tan²θ)

f(tan θ) = (1-sin²θ/cos²θ)(1+sin²θ/cos²θ)

f(tan θ) = (cos²θ-sin²θ)(cos²θ+sin²θ)

Now, the value of cos²θ+sin²θ is 1, So

f(tan θ) = (cos²θ-sin²θ)

Now, cos2θ = (cos²θ-sin²θ)

f(tan θ) = cos2θ

Hence, the value of f(tan θ) is cos2θ.

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