Question

If x+1/x=2.cos teeta then xpower n+1/x power n =

Answer 1

Answer:

$Put x=cosα+isinα=e </p><p>iα</p><p>$

$So </p><p>x</p><p>1</p><p> </p><p> =x </p><p>−1</p><p> =e </p><p>−iα</p><p> =cosα−isinα$

$Clearly x+ </p><p>x</p><p>1</p><p> </p><p> =2cosα$

$Hence x </p><p>n</p><p> + </p><p>x </p><p>n</p><p> </p><p>1</p><p> </p><p> =x </p><p>n</p><p> +x </p><p>−n</p><p> =(cosα+isinα) </p><p>n</p><p> +(cosα+isinα) </p><p>−n$

$=cosnα+isinnα+sinnα−isinnα</p><p> =2cosnα, Use Demoivre's theorem$