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Trignometry
30 Nov 2011 12:48:56 IST
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cot16cot44+cot44cot76-cot76cot16
= (cos16cos44)/(sin16sin44) + (cos44cos76)/(sin44sin76) - (cos76cos16)/(sin76sin76)
= (cos16cos44sin76 + cos44cos76sin16 - cos76cos16sin44) / (sin16sin44sin76)............i)
let me first simplify the numerator
cos16cos44sin76 + cos44cos76sin16 - cos76cos16sin44
= cos44(sin76cos16+sin16cos76) - cos76cos16sin44
= cos44sin92 - cos76cos16sin44 {using sinAcosB+sinBcosA= son(A+B)}
= 2sin46cos46cos44 - cos76cos16sin44 { sin92=2sin46cos46}
= 2sin46sin44cos44 - cos76cos16sin44
= sin44( 2sin46cos44 - cos76cos16)
= sin44[ (sin90 + sin2) - (cos92 + cos60)/2] {using formulas of 2sinAcosB and 2cosAcosB resp.}
= sin44[ (3/2-3cos92)/2] {sin2= -cos92}
= 3sin44[(cos60-cos92)/2]
= 3sin44sin16sin76 { since (cos60-cos92) = cos(76-16)-cos(76+16)= 2sin16sin76}
substituting this value in the place of numerator in i) we get
(3sin44sin16sin76)/(sin16sin44sin76)
= 3 ans. __/\__