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        sina/2+sinb/2+sinc/2=1+4sinpie-a/4.sinpie-b/4.sinpie-c/4
one year ago

kkbisht
90 Points
							You have not mentioned the condition : a+b+c=$\pi$=180LHS= Sin a/2  + sin b/2 + sin c/2  -1(Now usr T-identity   sinC + sinD = 2sin(C+D)/2)cos(C-D)/2) =>   2 sin((a+b)/4) cos((a-b)/4) + sin c/2  -1 => 2 sin ((a+b)/4) cos((a-b)/4) + sin c/2 -sin π/2{again apply formula on last two terms sinC - sinD = 2 cosA sinB }=> 2sin ((a+b)/4) cos((a-b)/4) + 2cos((π+c)/4)sin((c-π)/4){now a+b+c = π  so a+b=π-c}=> 2sin((π-c)/4) cos((a-b)/4) + 2cos((π+c)/4) sin(( c-π)/4)=>2sin((π-c)/4){ cos((a-b)/4) - cos((π+c)/4)} {now again use formula cos(A-B) - cos(A+B) = 2 sin((C+D)/2) sin ((D-C)/2)}=> 2sin((π-c)/4){ 2 sin ((a-b+π+c)/8) sin((π+c-a+b)/8)}{now use a+b+c=π in the last two terms}=>2sin((π-c)/4) { 2 sin ((2π-2b)/8) sin (( 2π- 2a)/8)} now after simplify all calculation, we will get=> 4 sin ((π-c)/4) sin((π-b)/4)sin((π-a)/4)therefore Sin a/2  + sin b/2 + sin c/2  -1=4 sin ((π-c)/4) sin((π-b)/4)sin((π-a)/4)Hence ProvedBykkbisht

one year ago
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• 31 Video Lectures
• Revision Notes
• Test paper with Video Solution
• Mind Map
• Study Planner
• NCERT Solutions
• Discussion Forum
• Previous Year Exam Questions