If in triangle ABC , C = 90 degrees , prove that : cosec (A-B) = a^2+b^2/a^2-b^2 and sec(A-B) = c^2/2ab .

Asked by Sanjay | 13th May, 2015, 12:36: AM

Expert Answer:

C o n s i d e r space L. H. S L. H. S equals cos e c open parentheses A minus B close parentheses space space space space space space space space space space equals fraction numerator 1 over denominator sin open parentheses A minus B close parentheses end fraction space space space space space space space space space space equals fraction numerator 1 over denominator sin A cos B minus cos A sin B end fraction.... left parenthesis 1 right parenthesis C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
T h u s comma space sin A equals a over c comma space cos B equals a over c S i m i l a r l y comma space cos A equals b over c comma space sin B equals b over c T h u s comma space e q u a t i o n space left parenthesis 1 right parenthesis space c a n space b e space w r i t t e n space a s comma cos e c open parentheses A minus B close parentheses equals fraction numerator 1 over denominator a over c cross times a over c minus b over c cross times b over c end fraction rightwards double arrow cos e c open parentheses A minus B close parentheses equals fraction numerator 1 over denominator a squared over c squared minus b squared over c squared end fraction rightwards double arrow cos e c open parentheses A minus B close parentheses equals fraction numerator 1 over denominator fraction numerator a squared minus b squared over denominator c squared end fraction end fraction rightwards double arrow cos e c open parentheses A minus B close parentheses equals fraction numerator c squared over denominator a squared minus b squared end fraction rightwards double arrow cos e c open parentheses A minus B close parentheses equals fraction numerator a squared plus b squared over denominator a squared minus b squared end fraction H e n c e space p r o v e d. N o w space l e space u s space c o n s i d e r space s e c open parentheses A minus B close parentheses colon s e c open parentheses A minus B close parentheses equals fraction numerator 1 over denominator cos open parentheses A minus B close parentheses end fraction rightwards double arrow s e c open parentheses A minus B close parentheses equals fraction numerator 1 over denominator cos A cos B plus sin A sin B end fraction rightwards double arrow s e c open parentheses A minus B close parentheses equals fraction numerator 1 over denominator b over c cross times a over c plus a over c cross times b over c end fraction rightwards double arrow s e c open parentheses A minus B close parentheses equals fraction numerator 1 over denominator fraction numerator a b over denominator c squared end fraction plus fraction numerator a b over denominator c squared end fraction end fraction rightwards double arrow s e c open parentheses A minus B close parentheses equals fraction numerator 1 over denominator fraction numerator 2 a b over denominator c squared end fraction end fraction rightwards double arrow s e c open parentheses A minus B close parentheses equals fraction numerator c squared over denominator 2 a b end fraction H e n c e space p r o v e d.

Answered by Vimala Ramamurthy | 13th May, 2015, 10:26: AM

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