If f(x) = \(\rm {2x\over {1\ +\ x^2}}\), then find the value of f(tan θ).

  1. sin 2θ
  2. cos 2θ
  3. tan 2θ
  4. sin θ

Answer (Detailed Solution Below)

Option 1 : sin 2θ
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Detailed Solution

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Concept:

Trigonometric Formulae:

\(\rm \sin 2x = \frac{2\tan x}{1\ +\ \tan^2x}\)

\(\rm \cos 2x = \frac{1\ -\ \tan^2x}{1\ +\ \tan^2x}\)

\(\rm \tan 2x = \frac{2\tan x}{1\ -\ \tan^2x}\)

Calculation:

We have f(x) = \(\rm {2x\over {1\ +\ x^2}}\).

Substituting x = tan θ, we get:

⇒ f(tan θ) = \(\rm {2\tan\theta\over {1\ +\ \tan^2 \theta}}\) = sin 2θ.

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