Question

prove that by trigonometric identity ​

Answer 1

$LHS, \\ \frac{ \cos(x) - 2 { (\cos(x) )}^{3} }{2( { \sin(x) ) ) }^{3} - \sin(x) } \\ taking \: common \: \cos(x) \: from \: {n}^{r} and \: \sin(x) \: from \: {d}^{r} \\ = \frac{ \cos(x)( 1 - 2 { (\cos(x) )}^{2} )}{ \sin(x)( { 2(\sin(x)) }^{2} - 1) } \\ we \: know \: \cos(2x) = 2 {( \cos(x)) }^{2} - 1 = 1 - 2 { (\sin(x) )}^{2} \\ by \: using \: above \: identity \\ = \frac{ \cos(x) ( - \cos(2x) )}{ \sin(x)( - \cos(2x) )} \\ = \cot(x) \\ RHS$

i wrote theta as 'x' because theta is not present in my keyboard. hope u understand.