Principle of Mathematical Induction

Principle of Mathematical Induction

Question

Find the general solution of:

cos space mx space plus space cos space nx space equals space 0






Answer
cos space mx space plus space cos space nx space equals space 0
rightwards double arrow space space 2 space cos open parentheses fraction numerator mx plus nx over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator mx space minus space nx over denominator 2 end fraction close parentheses space equals space 0     open square brackets because space cosC space plus space cosD space equals space 2 cos open parentheses fraction numerator straight C plus straight D over denominator 2 end fraction close parentheses space cos open parentheses fraction numerator straight C minus straight D over denominator 2 end fraction close parentheses close square brackets

rightwards double arrow  either space space cos open parentheses fraction numerator straight m plus straight n over denominator 2 end fraction close parentheses straight x space equals space 0                           or         cos open parentheses fraction numerator straight m minus straight n over denominator 2 end fraction close parentheses straight x space equals space 0

rightwards double arrow  either open parentheses fraction numerator straight m plus straight n over denominator 2 end fraction close parentheses straight x space equals space left parenthesis 2 straight p plus 1 right parenthesis straight pi over 2 comma space straight p space element of space straight Z         or      space space space open parentheses fraction numerator straight m minus straight n over denominator 2 end fraction close parentheses straight x space equals space left parenthesis 2 straight p plus 1 right parenthesis straight pi over 2 comma space space straight p space element of space straight Z

rightwards double arrow either straight x space equals space open parentheses fraction numerator 2 straight p plus 1 over denominator straight m plus straight n end fraction close parentheses straight pi comma space space straight p space element of space straight Z                     or      space space space straight x space equals space open parentheses fraction numerator 2 straight p plus 1 over denominator straight m minus straight n end fraction close parentheses straight pi comma space straight p element of space straight z

rightwards double arrow   straight x space equals space open parentheses fraction numerator 2 straight p plus 1 over denominator straight m plus straight n end fraction close parentheses straight pi                                       or       straight x space equals space open parentheses fraction numerator 2 straight p plus 1 over denominator straight m minus straight n end fraction close parentheses straight pi comma space space straight p space element of space straight Z

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