Let f(x)=2008x+2008-x/2, g(x)= 2008x-2008-x/2 then prove that f(x+y)=f(x)f(y)+g(x)g(y).

Asked by Anil | 11th May, 2017, 12:13: PM

Expert Answer:

begin mathsize 16px style straight f open parentheses straight x close parentheses equals fraction numerator 2008 to the power of straight x plus 2008 to the power of negative straight x end exponent over denominator 2 end fraction rightwards double arrow straight f open parentheses straight y close parentheses equals fraction numerator 2008 to the power of straight y plus 2008 to the power of negative straight y end exponent over denominator 2 end fraction
straight g open parentheses straight x close parentheses equals fraction numerator 2008 to the power of straight x minus 2008 to the power of negative straight x end exponent over denominator 2 end fraction rightwards double arrow straight g open parentheses straight y close parentheses equals fraction numerator 2008 to the power of straight y minus 2008 to the power of negative straight y end exponent over denominator 2 end fraction
Let comma space 2008 equals straight a
RHS equals straight f open parentheses straight x close parentheses straight f open parentheses straight y close parentheses plus straight g open parentheses straight x close parentheses straight g open parentheses straight y close parentheses
equals fraction numerator straight a to the power of straight x plus straight a to the power of negative straight x end exponent over denominator 2 end fraction cross times fraction numerator straight a to the power of straight y plus straight a to the power of negative straight y end exponent over denominator 2 end fraction plus fraction numerator straight a to the power of straight x minus straight a to the power of negative straight x end exponent over denominator 2 end fraction cross times fraction numerator straight a to the power of straight y minus straight a to the power of negative straight y end exponent over denominator 2 end fraction
equals fraction numerator straight a to the power of 2 straight x end exponent plus 1 over denominator 2 straight a to the power of straight x end fraction cross times fraction numerator straight a to the power of 2 straight y end exponent plus 1 over denominator 2 straight a to the power of straight y end fraction plus fraction numerator straight a to the power of 2 straight x end exponent minus 1 over denominator 2 straight a to the power of straight x end fraction cross times fraction numerator straight a to the power of 2 straight y end exponent minus 1 over denominator 2 straight a to the power of straight y end fraction
equals 1 half open parentheses fraction numerator open parentheses straight a to the power of 2 straight x end exponent plus 1 close parentheses open parentheses straight a to the power of 2 straight y end exponent plus 1 close parentheses over denominator straight a to the power of straight x plus straight y end exponent end fraction plus fraction numerator open parentheses straight a to the power of 2 straight x end exponent minus 1 close parentheses open parentheses straight a to the power of 2 straight y end exponent minus 1 close parentheses over denominator straight a to the power of straight x plus straight y end exponent end fraction close parentheses
after space solving space we space get
equals fraction numerator straight a to the power of straight x plus straight y end exponent plus straight a to the power of negative open parentheses straight x plus straight y close parentheses end exponent over denominator 2 end fraction
equals straight f open parentheses straight x plus straight y close parentheses
equals LHS
end style

Answered by Sneha shidid | 11th May, 2017, 05:02: PM