Question

Sin(a+b)-Sin(a-b)/Cos(a+b)-Cos(a-b)=Tan​

Answer 1

Answer:

Formula used -

sin a +sin b = 2sin (a+b)/2 cos(a-b) /2

cos a - cos b = -2sin(a+b)/2 sin(a-b) /2

$\frac{ \sin( \alpha + \beta ) - \sin( \alpha - \beta ) }{ \cos( \alpha + \beta ) - \cos( \alpha - \beta ) } \\ = \frac{ 2 \sin( \frac{ (\alpha + \beta ) + ( \alpha - \beta )}{2} ) \cos( \frac{( \alpha + \beta ) - ( \alpha - \beta )}{2} ) }{ - 2 \sin( \frac{( \alpha + \beta ) + ( \alpha - \beta )}{2} ) \sin( \frac{ \alpha + \beta ) - ( \alpha - \beta ) }{2} ) } \\ = \frac{\sin( \frac{ \alpha + \beta + \alpha - \beta }{2} ) \cos( \alpha + \beta - \alpha + \beta ) }{ - \sin( \frac{ \alpha + \beta + \alpha - \beta }{2} ) \sin( \frac{ \alpha + \beta - \alpha + \beta }{2} ) } \\ = \frac{ \cos( \frac{2 \beta }{2} ) }{ - \sin( \frac{2 \beta }{2} ) } \\ = - \cot( \frac{2 \beta }{2} ) \\ = - \cot( \beta )$

Answer 2

Answer:-

=> - cot(β)

Further Explanation

Formula used to do this question:-

$sin a +sin b = 2sin (a+b)/2 cos(a-b) /2$

$cos a - cos b </p><p>= -2sin(a+b)/2 sin(a-b) /2$

So,

$=>\begin{lgathered}\frac{ \sin( \alpha + \beta ) - \sin( \alpha - \beta ) }{ \cos( \alpha + \beta ) - \cos( \alpha - \beta ) } \\ \\=> \frac{ 2 \sin( \frac{ (\alpha + \beta ) + ( \alpha - \beta )}{2} ) \cos( \frac{( \alpha + \beta ) - ( \alpha - \beta )}{2} ) }{ - 2 \sin( \frac{( \alpha + \beta ) + ( \alpha - \beta )}{2} ) \sin( \frac{ \alpha + \beta ) - ( \alpha - \beta ) }{2} ) } \\ \\\frac{\sin( \frac{ \alpha + \beta + \alpha - \beta }{2} ) \cos( \alpha + \beta - \alpha + \beta ) }{ - \sin( \frac{ \alpha + \beta + \alpha - \beta }{2} ) \sin( \frac{ \alpha + \beta - \alpha + \beta }{2} ) } \\\\ => \frac{ \cos( \frac{2 \beta }{2} ) }{ - \sin( \frac{2 \beta }{2} ) } \\ => - \cot( \frac{2 \beta }{2} ) \\\\ => - \cot( \beta )\end{lgathered}$