Principle of Mathematical Induction

Principle of Mathematical Induction

Question

If tan left parenthesis straight alpha plus straight theta right parenthesis space equals space straight n space tan left parenthesis straight alpha minus straight theta right parenthesis comma show that left parenthesis straight n plus 1 right parenthesis space sin 2 straight theta space equals space left parenthesis straight n minus 1 right parenthesis space sin 2 straight alpha

Answer
tan space left parenthesis straight alpha plus straight theta right parenthesis space equals space straight n space tan left parenthesis straight alpha minus straight theta right parenthesis
rightwards double arrow        space space space space space fraction numerator sin space left parenthesis straight alpha plus straight theta right parenthesis over denominator cos space left parenthesis straight alpha plus straight theta right parenthesis end fraction space equals space fraction numerator nsin space left parenthesis straight alpha minus straight theta right parenthesis over denominator cos space left parenthesis straight alpha minus straight theta right parenthesis end fraction

Applying componendo and dividendo, we get
fraction numerator sin left parenthesis straight alpha plus straight theta right parenthesis space cos left parenthesis straight alpha minus straight theta right parenthesis space plus space cos left parenthesis straight alpha plus straight theta right parenthesis space sin left parenthesis straight alpha minus straight theta right parenthesis over denominator sin left parenthesis straight alpha plus straight theta right parenthesis space cos left parenthesis straight alpha minus straight theta right parenthesis space minus space cos left parenthesis straight alpha plus straight theta right parenthesis space sin left parenthesis straight alpha minus straight theta right parenthesis end fraction space equals space fraction numerator straight n plus 1 over denominator straight n minus 1 end fraction

rightwards double arrow          fraction numerator sin space left parenthesis straight alpha plus straight theta plus straight alpha minus straight theta right parenthesis over denominator sin space left parenthesis straight alpha plus straight theta minus straight alpha plus straight theta right parenthesis end fraction space equals space fraction numerator straight n plus 1 over denominator straight n minus 1 end fraction

rightwards double arrow                space space space fraction numerator sin 2 straight alpha over denominator sin 2 straight theta end fraction space equals space fraction numerator straight n plus 1 over denominator straight n minus 1 end fraction
rightwards double arrow         left parenthesis straight n minus 1 right parenthesis space sin 2 straight alpha space equals space left parenthesis straight n plus 1 right parenthesis space sin space 2 straight theta
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