Question

prove that (1/cosecA-cotA)-(1/sinA)=(1/sinA)-(1/cosecA+cotA)

Answer 1

LHS = 1/cosec A - cot A - 1/ sin A

       = 1/cosec A - cot A * cosec A + cot A/cosec A + cot A - 1/sin A

       = (cosec A + cot A)/(cosec^2A - cot^2A) - 1/sin A

      = (cosec A + cot A)/(cosec^2A - cot^2A) - cosec A

      = cosec A + cot A - cosec A

      = cot A.


RHS = 1/sin A - 1/cosecA+cotA

        = cosecA - (cosecA - cotA)/(cosec^2A - cot^2A)

       = cosec A - (cosec A - cot A)

       = cosec A - cosec A + cot A

       = cot A.


LHS = RHS.


Hope this helps!

Answer 2

hello users ........

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we have to prove that:

$\frac{1}{cosec A - cot A } - \frac{1}{sin A}$ = $\frac{1}{sin A } - \frac{1}{cosec A + cot A}$

solution :-
Taking LHS:
$\frac{1}{cosec A - cot A } - \frac{1}{sin A}$ 

= $\frac{1*(cosec A + cot A)}{( cosec A- cot A )*(cosec A + cot A)} - cosec A$

= $\frac{(cosec A + cot A)}{( cosec^{2} A- cot^{2} A)} - cosec A$

= ( cosec A + cot A ) - cosec A     .........( cosec²x - cot²x = 1)

= cot A 

now 
taking RHS
 $\frac{1}{sin A } - \frac{1}{cosec A + cot A}$

=$cosec A - \frac{1*(cosec A - cot A )}{( cosec A + cot A ) * ( cosec A - cot A)}$

= $cosec A - \frac{(cosecA - cot A)}{(cosec^{2}A - cot^{2}A ) }$

= cosec A - ( cosec A - cot A ) / 1    .... (cosec²x - cot²x = 1)

= cot A 

Hence 
LHS = RHS 

Proved ........

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✪✪ hope it helps ✪✪