Question

prove that
1/(cosec a - cot a) - 1/sin a = 1/tan a

Answer 1

$\implies\sf{\dfrac{1}{cosecA-cotA } -\dfrac{1}{sinA}}\\\\\\\implies\sf{\dfrac{1}{cosecA-cotA} \times \dfrac{cosecA+cotA}{cosecA+cotA} -\dfrac{1}{sinA} }\\\\\\\implies\sf{\dfrac{cosecA+cotA}{(cosecA-cotA)(cosecA-cotA)}-\dfrac{1}{sinA} }\\\\\\\implies\sf{\dfrac{cosecA+cotA}{cosec^2 A -cot^2A } -\dfrac{1}{sinA} }$

  Using:

           cosec²x - cot²x = 1   ;

           1/sinx = cosecx

           cotx = 1/tanx

$\implies\sf{\dfrac{cosecA+cotA}{1}- cosecA}\\\\\implies \sf{cosecA+cotA-cosecA }\\\\\implies\sf{cotA }\\\\\implies\sf{\dfrac{1}{tanA} }$

Answer 2

Refer the given attachment above

Hope this will help you Buddy