Question

tan/1-cot+cot/1-tan=1+tan+cos​

Answer 1

$\orange{ \underline{\underline{correct \: question}}} \\ \green{\frac{ \tan(o) }{1 - \cot(o)} + \frac{ \cot(o) }{1 - \tan(o) }} \\ \green{ = 1 + \sec(o) \csc(o)} \\ \green{o = theta} \\ \mathcal{\green{LHS}} \\ {\frac{ \frac{\sin(o) }{\cos(o)}}{1 - \frac{\cos(o) }{\sin(o)}} + \frac{ \frac{ \cos(o)}{\sin(o) }}{1 - \frac{\sin(o)}{ \cos(o) }}} \\ {\frac{\frac{\sin(o) }{\cos(o) }}{\frac{\sin(o) - \cos(o) }{\sin(o) }} + \frac{\frac{\cos(o) }{\sin(o) }}{ \frac{ \cos(o) - \sin(o) }{ \cos(o) }}} \\ \\ = { \frac{\sin^{2} (o) }{\cos(o)(\sin(o) - \cos(o)}} \\ + {\frac{ \cos^{2} ( o ) }{ \sin(o)( \cos(o) - \sin(o))}} \\ \\ = { \frac{ \sin^{2} (o) }{ \cos(o)( \sin(o) - \cos(o))}} \: \: \\ - \: {\frac{ \cos^{2} (o) }{ \sin(o)( \sin(o) - \cos(o)}} \\ \\ = { \frac{ \sin ^{3} (o) - \cos ^{3} (o) }{ \cos(o) \sin(o)( \sin(o) - \cos(o)) }} \\ \\ = { \frac{( \sin(o) - \cos(o)( \sin^{2} (o) + \cos^{2} (o) + \sin(o) \cos(o) )}{ \cos(o) \sin(o)( \sin(o) - \cos(o) )}} \\ \\ = { \frac{1 + \sin(o) \cos(o) }{ \cos(o) \sin(o)}} \\ \\ = { \frac{1}{ \cos(o) \ \sin(o)} + \frac{\sin(o) \cos(o)}{\cos(o) \sin(o)}} \\ \\ \orange{\underline{\underline{using \: commonando \: and \: dividendo}}} \\ \green{ \underline{\underline{\sec(o) + \csc(o) + 1 \: = \: \: proved}}} \\ \\ \orange{ \underline{\underline{LHS = RHS}}}$