If fx -2 x 1- x 2- show that ftan-sin 2 -

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Standard XII

Mathematics

Parametric Differentiation

If fx =2 x 1+...

Question

If $f (x) = \frac{2 x}{1 + x^{2}}$ , show that f(tan θ) = sin 2θ.

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Solution

Given:
$f (x) = \frac{2 x}{1 + x^{2}}$
Thus,
$f (\tan θ) = \frac{2 (\tan θ)}{1 + \tan^{2} θ}$
$= \frac{2 \times \frac{\sin θ}{\cos θ}}{1 + (\frac{\sin^{2} θ}{\cos^{2} θ})} = \frac{2 \sin θ}{\cos θ} \times \frac{\cos^{2} θ}{\cos^{2} θ + \sin^{2} θ} = \frac{2 \sin θ \cos θ}{1} [∵ \cos^{2} θ + \sin^{2} θ = 1] = \sin 2 θ [∵ 2 \sin θ \cos θ = \sin 2 θ]$

Hence, f (tan θ) = sin 2θ.

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If fx=2x1+x2, show that f(tan θ) = sin 2θ.

Given: fx=2x1+x2 Thus, ftanθ=2tanθ1+tan2θ =2×sin θcos θ1+sin2θcos2θ= 2 sin θ cos θ×cos2θcos2θ+sin2θ= 2 sin θ cos θ1 ∵cos2θ+sin2θ =1=sin 2θ ∵ 2 sin θ cos θ = sin 2θ Hence, f (tan θ) = sin 2θ.

If $f (x) = \frac{2 x}{1 + x^{2}}$ , show that f(tan θ) = sin 2θ.