Question

please give me right answers with full method please. urgently​

Answer 1

To simplify :

$\star \: { \tan}^{ - 1} ( \frac{3 {a}^{2}x - {x}^{3} }{ {a}^{3} - 3a {x}^{2} } )$

Here ,

$\star \: let \: x = a \tan \theta \\ \\ \implies \: \frac{x}{a} = \tan \theta \: \\ \\ \implies \theta = { \tan}^{ - 1} ( \frac{x}{a} )$

$\implies \: { \tan}^{ - 1} ( \frac{3 {a}^{2} (a \tan \theta) - ( { a \tan \theta)}^{3} }{ {a}^{3} \: - 3a { (a \tan \theta)}^{2} } ) \\ \\ \implies \: { \tan}^{ - 1} ( \frac{ \cancel{{a}^{3}}(3 \tan \theta - { \tan}^{3} \theta)}{ \cancel{ {a}^{3}}(1 - 3 { \tan}^{2} \theta \: ) } ) \\ \\ \implies \: { \tan}^{ - 1} ( \tan3 \theta) \\ \\ \implies \: \: 3 \theta \\ \\ \implies \: 3 { \tan}^{ - 1} ( \frac{x}{a} )$

Formula used:

$\star \boxed{ \red{ \bold{ \tan \: 3 \theta = \frac{3 \tan \theta - { \tan}^{3} \theta }{1 - 3 { \tan}^{2} \theta } }}} \\ \\ \\ \star \boxed{ \bold{ \blue{{ \tan}^{ - 1} ( \tan \theta) = \theta}}}$

Answer 2

HERE IS U R ANSWER DEAR PLZ FOLLW ME