(cosec theta - cot theta ) sq. = 1- cos theta/ 1+ cos theta: prove that
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(cosec theta - cot theta ) sq. = 1- cos theta/ 1+ cos theta: prove that
Solution :
Take L.H.S = (cosecФ - cot Ф ) ^2
= (1/ sin Ф - cos Ф / sin Ф )^2
= (1 - cos Ф)^2 / sin^2 Ф
= ( 1 - cos Ф )( 1 - cos Ф ) / sin^2 Ф
= ( 1 - cos Ф )( 1 - cos Ф ) / 1 - cos ^2 Ф
Note :
sin^2 Ф + cos ^2 Ф = 1
=> sin^2 Ф = 1 - cos ^2 Ф
= ( 1 - cos Ф )( 1 - cos Ф ) / 1 - cos ^2 Ф
= ( 1 - cos Ф )( 1 - cos Ф ) / ( 1 - cos Ф ) (1+cos Ф)
Note :
( a )^2 - (b)^2 = (a +b )(a-b)
= (1- cos Ф) / (1+ cos Ф ) = R.H.S
Hope this is helpful!!!
Solution :
Take L.H.S = (cosecФ - cot Ф ) ^2
= (1/ sin Ф - cos Ф / sin Ф )^2
= (1 - cos Ф)^2 / sin^2 Ф
= ( 1 - cos Ф )( 1 - cos Ф ) / sin^2 Ф
= ( 1 - cos Ф )( 1 - cos Ф ) / 1 - cos ^2 Ф
Note :
sin^2 Ф + cos ^2 Ф = 1
=> sin^2 Ф = 1 - cos ^2 Ф
= ( 1 - cos Ф )( 1 - cos Ф ) / 1 - cos ^2 Ф
= ( 1 - cos Ф )( 1 - cos Ф ) / ( 1 - cos Ф ) (1+cos Ф)
Note :
( a )^2 - (b)^2 = (a +b )(a-b)
= (1- cos Ф) / (1+ cos Ф ) = R.H.S
Hope this is helpful!!!
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