Principle Of Mathematical Induction

Question
CBSEENMA11014287

tan left parenthesis straight x minus straight y right parenthesis space equals space fraction numerator tanx minus tany over denominator 1 plus tanx space tany end fraction comma space where space straight x comma straight y comma space straight x minus straight y not equal to left parenthesis 2 straight n plus 1 right parenthesis straight pi over 2 comma space straight n element of straight Z

Solution

As we know that:
tan space left parenthesis straight x plus straight y right parenthesis space equals space fraction numerator tanx plus tany over denominator 1 minus tanx space tany end fraction
Replacing y by -y, we get
                  tan left parenthesis straight x minus straight y right parenthesis space equals space fraction numerator tanx plus tan left parenthesis negative straight y right parenthesis over denominator 1 minus tanx space tan left parenthesis negative straight y right parenthesis end fraction space equals space fraction numerator tanx minus tany over denominator 1 plus tanx space tany end fraction   [∵ tan (-y) = -tany]
Hence,   tan left parenthesis straight x minus straight y right parenthesis equals fraction numerator tanx minus tany over denominator 1 plus tanx space tany end fraction

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