Question

This Question is so much hard so solve it.​

Answer 1

To find :

Value of

$\sf \: \: \dfrac{ \tan( \theta) }{ \dfrac{ { \sin^3( \theta) } }{ \cos( \theta) } + ( \sin( \theta) \times \cos( \theta) )}$

Identity used :

$\sf \bullet \: \sin^2( x) + { \cos^2(x) } = 1 \\ \\ \sf \bullet \: \: \dfrac{ \sin(x) }{ \cos(x) } = \tan(x)$

Solution :

$\sf \implies \: \dfrac{ \tan( \theta) }{ \dfrac{ { \sin^3( \theta) } }{ \cos( \theta) } + ( \sin( \theta) \times \cos( \theta) )} \\ \\ \sf \: take \: lcm \: in \: denominator \\ \sf \implies \: \dfrac{ \tan( \theta) }{ \dfrac{ { \sin^3( \theta) } \: + \{ \: \cos( \theta) \: \times ( \: \sin( \theta) \times \cos( \theta) ) \} }{ \cos( \theta) } } \\ \\ \sf \implies \: \dfrac{ \tan( \theta) }{ \dfrac{ { \sin^3( \theta) } \: + \{ \: \: \sin( \theta) \times \cos^2( \theta) \} }{ \cos( \theta) } } \\ \\ \sf \: take \: \sin( \theta ) \: common \\ \\ \sf \implies \: \dfrac{ \tan( \theta) }{ \dfrac{ \sin( \theta) \times \: \{ \: \: \sin^2( \theta) + \: \cos^2( \theta) \} }{ \cos( \theta) } } \\ \\ \sf \implies \: \dfrac{ \tan( \theta) }{ \dfrac{ \sin( \theta) \times \: \{ \: 1\} }{ \cos( \theta) } } \\ \\ \sf \implies \: \dfrac{ \tan( \theta)}{ \dfrac{ \sin( \theta) }{ \cos( \theta) } } \\ \\ \sf \implies \: \dfrac{ \tan( \theta) }{ \tan( \theta) } \\ \\ \sf \implies \: 1$

ANSWER : 1