Principle of Mathematical Induction

Principle of Mathematical Induction

Question

If space space tanx space equals space straight b over straight a comma find the value of straight a space cos space 2 straight x space plus space straight b space sin space 2 straight x.

Answer

Using   space space cos space 2 straight x space equals space fraction numerator 1 minus tan squared straight x over denominator 1 plus tan squared straight x end fraction  and
            sin 2 straight x space equals space fraction numerator 2 tanx over denominator 1 plus tan squared straight x end fraction comma we get
space space space straight a space cos space 2 straight x plus straight b space sin space 2 straight x equals straight a open square brackets fraction numerator 1 minus tan squared straight x over denominator 1 plus tan squared straight x end fraction close square brackets plus straight b open square brackets fraction numerator 2 tanx over denominator 1 plus tan squared straight x end fraction close square brackets space equals space straight a open square brackets fraction numerator 1 minus begin display style straight b squared over straight a squared end style over denominator 1 plus begin display style straight b squared over straight a squared end style end fraction close square brackets plus straight b open square brackets fraction numerator 2 open parentheses begin display style straight b over straight a end style close parentheses over denominator 1 plus begin display style straight b squared over straight a squared end style end fraction close square brackets
                            space space space equals straight a open square brackets fraction numerator straight a squared minus straight b squared over denominator straight a squared plus straight b squared end fraction close square brackets plus straight b open square brackets fraction numerator 2 ab over denominator straight a squared plus straight b squared end fraction close square brackets space equals space fraction numerator 1 over denominator straight a squared plus straight b squared end fraction left square bracket straight a cubed minus ab squared plus 2 ab squared right square bracket
                               equals fraction numerator 1 over denominator straight a squared plus straight b squared end fraction left square bracket straight a cubed plus ab squared right square bracket space equals space fraction numerator 1 over denominator straight a squared plus straight b squared end fraction cross times space straight a space left parenthesis straight a squared plus straight b squared right parenthesis space equals space straight a
Hence,       straight a space cos 2 straight x space plus space straight b space sin space 2 straight x space equals space straight a.

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