\(\tan ^{-1}\left(\frac{a}{b}\right)-\tan ^{-1}\left(\frac{a-b}{a+b}\right)\) किसके बराबर है?

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  1. \(-\frac{\pi}{4}\)
  2. \(\frac{\pi}{4}\)
  3. \(\tan ^{-1}\left(\frac{a^2-b^2}{a^2+b^2}\right)\)
  4. \(\tan ^{-1}\left(\frac{2 a b}{a^2+b^2}\right)\)

Answer (Detailed Solution Below)

Option 2 : \(\frac{\pi}{4}\)
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अवधारणा:

सूत्र: tan-1x - tan-1y = tan-1 \(x-y\over 1+xy\)

स्पष्टीकरण:

\(\tan ^{-1}\left(\frac{a}{b}\right)-\tan ^{-1}\left(\frac{a-b}{a+b}\right)\)

= \(\tan ^{-1}\left({\frac{a}{b}-{a-b\over a+b}\over1+\frac ab.{a-b\over a+b}}\right)\)

= \(\tan ^{-1}\left({\frac{a^2+ab-ab+b^2}{b(a+b)}\over{ab+b^2+a^2-ab\over b(a+b)}}\right)\)

= tan -1 \(a^2+b^2\over a^2+b^2\) = tan -1 1 = \(\frac{\pi}{4}\)

विकल्प (2) सत्य है।

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