A) \[\frac{{{m}^{2}}+{{n}^{2}}}{{{m}^{2}}-{{n}^{2}}}\]
B) \[\frac{2\,\,mn}{{{m}^{2}}+{{n}^{2}}}\]
C) \[\frac{{{m}^{2}}-{{n}^{2}}}{2mn}\]
D) \[\frac{{{m}^{2}}+{{n}^{2}}}{2\,\,mn}\]
Correct Answer: C
Solution :
Given, \[\sin \theta =\frac{{{m}^{2}}-{{n}^{2}}}{{{m}^{2}}+{{n}^{2}}}\] In \[\Delta ABC,\] \[AB=\sqrt{{{(AC)}^{2}}-{{(BC)}^{2}}}\] \[=\sqrt{{{m}^{4}}+{{n}^{4}}+2{{m}^{2}}{{n}^{2}}-({{m}^{4}}+{{n}^{4}}-2{{m}^{2}}{{n}^{2}})}\] \[=\sqrt{4{{m}^{2}}{{n}^{2}}}=2mn\] \[\therefore \]\[\tan \theta =\frac{{{m}^{2}}-{{n}^{2}}}{2\,\,mn}\]
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