Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Show that:  

8 cos cubed open parentheses straight pi over 9 close parentheses minus 6 cos open parentheses straight pi over 9 close parentheses space equals space 1

Answer

L.H.S. = 8 cos cubed open parentheses straight pi over 9 close parentheses minus 6 cos open parentheses straight pi over 9 close parentheses space equals space 8 cos cubed straight A space minus space 6 space cosA comma where space space space straight A space equals space straight pi over 9
          space space space equals space 2 open square brackets 4 space cos cubed straight A space minus space 3 space cosA close square brackets space equals space 2 space cos space 3 straight A
           space space space equals space 2 space cos space 3 open parentheses straight pi over 9 close parentheses space equals space 2 space cos open parentheses straight pi over 3 close parentheses space equals space 2 cross times 1 half space equals space 1 space equals space space straight R. straight H. straight S.
∴        L.H.S. = R.H.S.
Hence,           8 cos cubed open parentheses straight pi over 9 close parentheses minus 6 cos open parentheses straight pi over 9 close parentheses equals 1

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