If tan theta +sin theta=m and …

Given, {tex}\tan \theta + \sin \theta = m{/tex}    and {tex}\tan \theta - \sin \theta = n{/tex}

L.H.S = {tex}m^{2}-n^{2}{/tex}

{tex}=(tan\theta+sin\theta)^{2}-(tan\theta-sin\theta)^{2}{/tex}

{tex}=tan^{2}\theta+sin^{2}\theta+2tan\theta sin\theta-[tan^{2}\theta+sin^{2}\theta-2tan\theta sin\theta]{/tex}

{tex}=tan^{2}\theta+sin^{2}\theta+2tan\theta sin\theta-tan^{2}\theta-sin^{2}\theta+2tan\theta sin\theta{/tex}

{tex}=4tan\theta sin\theta{/tex}

R.H.S = 4{tex}\sqrt{mn}{/tex}

{tex}=4\sqrt{(tan\theta+sin\theta)(tan\theta-sin\theta)}{/tex}

{tex}=4\sqrt{tan^{2}\theta-sin^{2}\theta}{/tex}

{tex}=4\sqrt{\frac{sin^{2}\theta}{cos^{2}\theta}-sin^{2}\theta}{/tex}

{tex}=4\sqrt{\frac{sin^{2}\theta-sin^{2}\theta cos^{2}\theta}{cos^{2}\theta}}{/tex}

{tex}=\frac{4}{cos\theta}\sqrt{sin^{2}\theta-sin^{2}\theta cos^{2}\theta}{/tex}

{tex}=\frac{4}{cos\theta}\sqrt{sin^{2}\theta (1-cos^{2}\theta)}{/tex}

{tex}=\frac{4}{cos\theta} \times sin\theta \times \sqrt{sin^{2}\theta}{/tex}

{tex}=4tan\theta sin\theta{/tex}

Hence, L.H.S = R.H.S