Question
Tue March 03, 2015 By: Krunal Badlani

IF COS X = COS(A/2)COS(B/2) / 1-SIN(A/2)SIN(B/2) . PROVE THAT TAN(X/2) = TAN(A/2)-TAN(B/2) / 1-TAN(A/2)TAN(B/2) .PLEASE ANSWER FAST I HAVE MY FINAL EXAM TOMORROW PLEASE

Expert Reply
Prasenjit Paul
Tue March 03, 2015
cos space x equals fraction numerator cos space begin display style A over 2 end style. cos begin display style B over 2 end style over denominator 1 minus sin space begin display style A over 2 end style. sin space begin display style B over 2 end style end fraction........ left parenthesis i right parenthesis 2 cos squared x over 2 minus 1 equals fraction numerator cos space begin display style A over 2 end style. cos begin display style B over 2 end style over denominator 1 minus sin space begin display style A over 2 end style. sin space begin display style B over 2 end style end fraction 2 cos squared x over 2 equals 1 plus fraction numerator cos space begin display style A over 2 end style. cos begin display style B over 2 end style over denominator 1 minus sin space begin display style A over 2 end style. sin space begin display style B over 2 end style end fraction 2 cos squared x over 2 equals fraction numerator 1 minus sin B over 2 sin A over 2 plus cos B over 2 cos A over 2 over denominator 1 minus sin B over 2 sin A over 2 end fraction 2 cos squared x over 2 equals fraction numerator 1 plus cos B over 2 cos A over 2 minus sin B over 2 sin A over 2 over denominator 1 minus sin B over 2 sin A over 2 end fraction 2 cos squared x over 2 equals fraction numerator 1 plus cos open parentheses begin display style fraction numerator A plus B over denominator 2 end fraction end style close parentheses over denominator 1 minus sin B over 2 sin A over 2 end fraction........ left parenthesis i i right parenthesis  1 minus 2 sin squared x over 2 equals fraction numerator cos space begin display style A over 2 end style. cos begin display style B over 2 end style over denominator 1 minus sin space begin display style A over 2 end style. sin space begin display style B over 2 end style end fraction space left square bracket F r o m space left parenthesis i right parenthesis right square bracket 2 sin squared x over 2 equals 1 minus fraction numerator cos space begin display style A over 2 end style. cos begin display style B over 2 end style over denominator 1 minus sin space begin display style A over 2 end style. sin space begin display style B over 2 end style end fraction 2 sin squared x over 2 equals fraction numerator 1 minus sin B over 2 sin A over 2 minus cos B over 2 cos A over 2 over denominator 1 minus sin B over 2 sin A over 2 end fraction 2 sin squared x over 2 equals fraction numerator 1 minus cos A over 2 cos B over 2 minus sin A over 2 sin B over 2 over denominator 1 minus sin B over 2 sin A over 2 end fraction 2 sin squared x over 2 equals fraction numerator 1 minus cos open parentheses begin display style fraction numerator A minus B over denominator 2 end fraction end style close parentheses over denominator 1 minus sin B over 2 sin A over 2 end fraction......... left parenthesis i i i right parenthesis left parenthesis i i i right parenthesis divided by left parenthesis i i right parenthesis fraction numerator sin squared x over 2 over denominator cos squared x over 2 end fraction equals fraction numerator 1 minus cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses over denominator 1 plus cos open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses end fraction tan squared x over 2 equals fraction numerator 2 sin squared open parentheses fraction numerator A minus B over denominator 4 end fraction close parentheses over denominator 2 cos squared open parentheses fraction numerator A plus B over denominator 4 end fraction close parentheses end fraction tan x over 2 equals fraction numerator sin open parentheses fraction numerator A minus B over denominator 4 end fraction close parentheses over denominator cos open parentheses fraction numerator A plus B over denominator 4 end fraction close parentheses end fraction space left square bracket T a k i n g space S q u a r e space r o o t space o n space b o t h space s i d e s right square bracket tan x over 2 equals fraction numerator sin A over 4 cos begin display style B over 4 end style minus cos begin display style A over 4 end style sin begin display style B over 4 end style over denominator cos A over 4 cos B over 4 minus sin A over 4 sin B over 4 end fraction tan x over 2 equals fraction numerator sin A over 4 cos begin display style B over 4 end style minus cos begin display style A over 4 end style sin begin display style B over 4 end style over denominator cos A over 4 cos B over 4 minus sin A over 4 sin B over 4 end fraction space left square bracket D i v i d i n g space b o t h space n u m e r a t o r space a n d space D e n o m i n a t o r space b y space cos A over 4 cos B over 4 right square bracket tan x over 2 equals fraction numerator begin display style tan A over 4 end style minus begin display style tan B over 4 end style over denominator 1 minus tan A over 4 tan B over 4 end fraction P l e a s e space c h e c k space t h e space q u e s t i o n space o n c e space a g a i n.
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