Principle of Mathematical Induction

Principle of Mathematical Induction

Question

If cos x = 1 half open parentheses straight a plus 1 over straight a close parentheses comma show that:

cos 3 straight x space equals space 1 half open parentheses straight a cubed plus 1 over straight a cubed close parentheses


Answer
cos 3 straight x space equals space 4 cos cubed straight x minus 3 cosx space equals space 4 open curly brackets 1 half open parentheses straight a plus 1 over straight a close parentheses close curly brackets cubed space minus space 3 over 2 open parentheses straight a plus 1 over straight a close parentheses
           equals 4 open square brackets 1 over 8 open curly brackets straight a cubed plus 1 over straight a cubed plus 3 open parentheses straight a plus 1 over straight a close parentheses close curly brackets minus 3 over 2 open parentheses straight a plus 1 over straight a close parentheses close square brackets

            equals 1 half open parentheses straight a cubed plus 1 over straight a cubed close parentheses plus 3 over 2 open parentheses straight a plus 1 over straight a close parentheses minus 3 over 2 open parentheses straight a plus 1 over straight a close parentheses space equals space 1 half open parentheses straight a cubed plus 1 over straight a cubed close parentheses
Hence,    cos 3 straight x space equals space 1 half open parentheses straight a cubed plus 1 over straight a cubed close parentheses

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