Question
Sun September 15, 2013 By: Aditya Chakravarty

prove that- i)cos?cosec?-sin?sec?/cos?+sin?=cosec?-sec? ii)cot^2?(sec?-1)/(1+sin?)=sec^2?(1-sin?)/(1+sec?) iii)(1-cos?)^1/2/(1+cos?)^1/2=sin?/1+cos? iv)(1+sin?)^1/2/(1-sin?)^1/2=cos?/1-sin?

Expert Reply
Sun September 15, 2013
Please ask one question at a time. The solution to your first question is as follows:

L.H.S.

= cos A . cosec A - sin A . sec A / cos A + sin A

= [(1 / sec A) . cosec A - (1 / cosec A) . sec A ] / [(1 / sec A) + (1 / cosec A) ]

= [cosec A / sec A - sec A / cosec A ] / [ cosec A + sec A / sec A . cosec A ]

= [cosec2 A - sec2 A / sec A . cosec A ] / [ cosec A + sec A / sec A . cosec A ]

= (cosec2 A - sec2 A) (sec A . cosec A) / sec A . cosec A (sec A + cosec A)

= (cosec A - sec A) (cosec A + sec A) / (sec A + cosec A)

= cosec A - sec A

= R.H.S.

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