Question

if 2tanβ+cotβ=tanα prove that cotβ=2tan(α-β)​

Answer 1

$\huge\mathtt\pink{Given}$

2 tan beta+ cot beta = tan alpha

$\huge\mathtt\pink{To\:Prove}$

cot$\beta$ = 2 tan ( alpha- beta)

$\huge\mathtt\pink{Solution}$

$2 \: \tan( \alpha - \beta ) = 2 \frac{ \tan( \alpha ) - \tan( \beta ) }{1 + \tan( \alpha ) . \tan( \beta ) }$

✧Using the formula of tan ( a - b )

$2( \frac{2 \tan( \beta ) + \cot( \beta ) - \tan( \beta ) }{1 + (2 \tan( \beta ) + \cot( \beta ) ) \tan( \beta ) })$

✧ Using tan alpha = 2 tan beta + cot beta

$2( \frac{ \tan( \beta ) + \cot( \beta ) }{1 + 2 { \tan( \beta ) }^{2} + 1 } )$

$\frac{2( \tan( \beta ) + \cot( \beta )) }{2 + { \tan( \beta ) }^{2} }$

$\frac{2( \tan( \beta ) + \frac{1}{ \tan( \beta ) } )}{2(1 + { \tan( \beta ) }^{2}) }$

$\frac{1}{ \tan( \beta ) }$

→ cot beta

LHS = RHS