Question:

If \(\tan \theta = \frac12\) and \(\tan \phi = \frac13\) then the value of \(\theta + \phi\) is:

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When using the tangent addition formula, always simplify the numerator and denominator separately before dividing to avoid mistakes.
  • \(\frac\pi6\)
  • \(\pi\)
  • 0
  • \(\frac\pi4\)
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The Correct Option is D

Solution and Explanation

We use the tangent addition formula:
\[ \tan(\theta + \phi) = \frac\tan \theta + \tan \phi1 - \tan \theta \tan \phi \] Substitute \(\tan \theta = \frac12\) and \(\tan \phi = \frac13\):
\[ \tan(\theta + \phi) = \frac\frac12 + \frac131 - \frac12 \frac13 \] Simplify numerator: \(\frac12 + \frac13 = \frac3 + 26 = \frac56\)
Simplify denominator: \(1 - \frac16 = \frac56\)
Thus: \(\tan(\theta + \phi) = \frac\frac56\frac56 = 1\)
Since \(\tan(\theta + \phi) = 1\), \(\theta + \phi = \frac\pi4\) (principal value).
Hence, the correct answer is (D) \(\frac\pi4\).
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