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Here is your answer :
R.H.S = 2 cosec A
L.H.S = [ tan A / ( 1 + sec A ) ] - [ tan A / ( 1 - sec A ) ]
Taking the L.C.M of denominator :
= [ tan A ( 1 - sec A ) - tan A ( 1 + sec A ) ] / [ ( 1 + sec A ) ( 1 - sin A ) ]
Taking tan A common in numerator,
= [ tan A ( 1 - sec A - 1 - sec A ) ] / [ ( 1 + sec A ) ( 1 - sec A ) ]
Using identity :
[ ( a + b ) ( a - b ) = a² - b² ]
= [ tan A ( -2 sec A ) ] / [ 1 - sec² A ] --- ( 1 )
Using identity :
=> sec² A - tan²A = 1
=> -tan² A = 1 - sec² A ------- ( 2 )
Plug the value of ( 2 ) in ( 1 ),
= [ -2 × tan A × sec A ] / ( - tan² A )
= ( 2 sec A ) / tan A
Using identity :
=> sec A = 1/cos A
And
=> tan A = sin A / cos A
= ( 2 / cos A ) / ( sin A / cos A )
= ( 2 / cos A ) × ( cos A / sin A )
= 2 / sin A
Using identity :
=> ( 1 / cosec A ) = sin A
= 2 / ( 1 / cosec A)
= 2 × cosec A
=> 2 cosec A = R.H.S
Proved
Hope it helps !!
R.H.S = 2 cosec A
L.H.S = [ tan A / ( 1 + sec A ) ] - [ tan A / ( 1 - sec A ) ]
Taking the L.C.M of denominator :
= [ tan A ( 1 - sec A ) - tan A ( 1 + sec A ) ] / [ ( 1 + sec A ) ( 1 - sin A ) ]
Taking tan A common in numerator,
= [ tan A ( 1 - sec A - 1 - sec A ) ] / [ ( 1 + sec A ) ( 1 - sec A ) ]
Using identity :
[ ( a + b ) ( a - b ) = a² - b² ]
= [ tan A ( -2 sec A ) ] / [ 1 - sec² A ] --- ( 1 )
Using identity :
=> sec² A - tan²A = 1
=> -tan² A = 1 - sec² A ------- ( 2 )
Plug the value of ( 2 ) in ( 1 ),
= [ -2 × tan A × sec A ] / ( - tan² A )
= ( 2 sec A ) / tan A
Using identity :
=> sec A = 1/cos A
And
=> tan A = sin A / cos A
= ( 2 / cos A ) / ( sin A / cos A )
= ( 2 / cos A ) × ( cos A / sin A )
= 2 / sin A
Using identity :
=> ( 1 / cosec A ) = sin A
= 2 / ( 1 / cosec A)
= 2 × cosec A
=> 2 cosec A = R.H.S
Proved
Hope it helps !!
Anonymous:
better u solve in copy
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