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if T n =sin n theta+cos n theta ,prove that 6T 10 -15T 8 +10T 6 -1=0

if Tn=sinntheta+cosn theta  ,prove that


6T10-15T8+10T6-1=0


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2 Answers

Saurabh Koranglekar
askIITians Faculty 10335 Points
3 years ago
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Vikas TU
14149 Points
3 years ago
Dear student 
T1=sinX+cosX; T2=1; T3=(sinX)^3+(cosX)^3, etc. Let s=sinX, c=cosX:
T10=s^10+c^10; T8=s^8+c^8; T6=s^6+c^6, so:
6s^10+6c^10-15s^8-15c^8+10s^6+10c^6-1=
6s^10+6(1-s^2)^5-15s^8-15(1-s^2)^4+10s^6+10(1-s^2)^3-1=
[6s^10+6(1-5s^2+10s^4-10s^6+5s^8-s^10)]+[-15s^8-15(1-4s^2+6s^4-4s^6+s^8)]+...
...[10s^6+10(1-3s^2+3s^4-s^6)]-1=
-30s^8+6(1-5s^2+10s^4-10s^6+5s^8)-15(1-4s^2+6s^4-4s^6)+10(1-3s^2+3s^4)-1=
-30s^8+30s^8+(6-15+10-1)+s^2(-30+60-30)+s^4(60-90+30)+s^6(-60+60)=0 QED
Good Luck 

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