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  • If tan θ + sin θ = m and tan θ – sin θ = n, then prove that m2 – n2 = 4 sin θ tan θ [Hint: m + n = 2tanθ, m – n = 2 sin θ, then use m2 – n2 = (m + n)(m – n)]

If \tan \theta + \sin \theta = m and \tan \theta - \sin \theta = n, then prove that m\textsuperscript{2} - n\textsuperscript{2} = 4 \sin\theta \tan\theta

[Hint: m + n = 2\tan \theta , m - n = 2 \sin \theta, then use [m\textsuperscript{2} - n^2] = (m + n)(m - n)]

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\\\tan \theta +\sin \theta =m~and \tan \theta - \sin \theta =n \\\\ L.H.S=m^{2} - n^{2}= \left( m+n \right) \left( m - n \right) \\\\ = \left[ \left( \tan \theta +\sin \theta \right) + \left( \tan \theta - \sin \theta \right) \right] .~ \left[ \left( \tan \theta +\sin \theta \right) - \left( \tan \theta - \sin \theta \right) \right] \\\\ = \left( \\tan \theta +\sin \theta +\tan \theta - \sin \theta \right) . \left( \\tan \theta +\sin \theta - \tan \theta +\sin \theta \right) \\\\ =2 \tan \theta . 2\sin \theta =4 \sin \theta \tan \theta =R.H.S \\\\

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