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\( \tan \left[\frac{1}{2} \cos ^{-1}\left(\frac{3}{5}\right)\right]+\tan \left[\frac{1}{2} \cos ^{-1}\left(\frac{4}{5}\right)\right] \) |
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Answer» tan [1/2 cos-1 (3/5)] + tan [1/2 (cos-1 4/5)] Let 1/2 cos-1 3/5 = α and 1/2 cos-1 4/5 = β ⇒ cos-1 3/5 = 2α and cos-1 4/5 = 2β ⇒ cos 2α = 3/5 and cos 2β = 4/5 ⇒ 2cos2α - 1 = 3/5 and 2cos2β - 1 = 4/5 ⇒ 2cos2α = 1 + 3/5 = 8/5 and 2cos2β = 1 + 4/5 = 9/5 ⇒ cos2α = 8/10 and cos2β = 9/10 ⇒ cos α = \(\sqrt{8/10}\) and cos β = \(\sqrt{9/10}\) ⇒ tan α = √2/√8 = \(\sqrt{1/4}\) = 1/2 and tan β = 1/√9 = 1/3 ⇒ 1/2 cos-1 3/5 = α = tan-1 1/2 and 1/2 cos-1 4/5 = β = tan-1 1/3 ∴ tan (1/2 cos-1 (3/5)) + tan (1/2 cos-1 (4/5)) = tan(tan-1 1/2) + tan(tan-1 1/3) = 1/2 + 1/3 = 5/6. |
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