Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Show that:

cos space 6 straight A space equals space 32 space cos to the power of 6 straight A space minus space 48 space cos to the power of 4 straight A space plus space 18 space cos squared straight A space minus 1

Answer

Using cos 2 straight x space equals space 2 space cos squared straight x minus 1  and cos 3 straight A space equals space 4 space cos cubed straight A space minus space 3 space cosA comma we get
           cos 6 straight A space equals space 2 cos squared 3 straight A space minus space 1
                    equals space 2 left square bracket 4 space cos cubed straight A space minus space 3 space cosA right square bracket squared minus 1
                    equals 2 left square bracket space 16 space cos to the power of 6 space straight A space plus space 9 space cos squared straight A space minus space 24 space cos to the power of 4 straight A right square bracket space minus space 1
                space space space space space equals 32 cos to the power of 6 straight A space minus space 48 cos to the power of 4 straight A space plus space 18 space cos squared straight A minus 1
Hence,    cos space 6 straight A space equals space 32 space cos to the power of 6 straight A space minus space 48 space cos to the power of 4 straight A space plus space 18 space cos squared straight A space minus 1.
                    

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