Question

if tan=12/13 show that 2sin×cos/cos2-sin2=312/25​

Answer 1

$\huge \tt \: answer : \\ \: \\ \sf \: given : \\ \\ tan = \frac{12}{13} = \frac{adjecent \: side \: }{opposite \: side} \\ \: \: \: \\ \\ \bf \: {AC }^{2} ={BC }^{2} +{AB }^{2} = \sqrt{ {13}^{2} + {12}^{2} } = \sqrt{144 + 169} = \sqrt{313} \\ \\ now \\ \\LHS \\ \\ \: AC = \sqrt{313} \\ BC = 13 \\ AB = 12 \\ \\ \: then \: \\ \tt \huge : \longmapsto \: \frac{2 \: sin \: \times \: cos}{ {cos}^{2} - {sin}^{2} } \\ \\ \tt \huge : \longmapsto \: \frac{2 \times \frac{13}{ \sqrt{313} } \times \frac{12}{ \sqrt{313} } }{ {( \frac{12}{ \sqrt{313} } )}^{2} - {( \frac{13}{ \sqrt{313} }) }^{2} } \\ \\ \tt \huge : \longmapsto \: \frac{ \frac{26}{ \sqrt{313} } \times \frac{12}{ \sqrt{313} } }{ \frac{144}{313} - \frac{169}{313} } \\ \\ \tt \huge : \longmapsto \: \frac{ \frac{312}{313} }{ \frac{25}{313} } \\ \\ \tt \huge : \longmapsto \: \frac{312}{25}$

Hence proved

LHS=RHS