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if 2cosA=x+1/x,2cosB=y+1/y then show that 2cos(A-B)=x/y+y/x

if 2cosA=x+1/x,2cosB=y+1/y then show that
2cos(A-B)=x/y+y/x
 

Grade:12

1 Answers

Arun
25750 Points
4 years ago
 
Solve these as quadratic equations in x & y so that you get x = cosA + i sinA & cosA – i sinA; 
y = cosB + i sinB & cosB – isinB
If x = cosA + i sinA then 1/x = cosA – i sinA ;
y = cosB + i sinB then 1/y  cosB – isinB
x + 1/x = 2cosA => cosA = 1/2(x + 1/x)
x – 1/x = 2i sinA => sinA = 1/2i(x – 1/x)
y + 1/y = 2cosB => cosB = 1/2(y + 1/y)
y – 1/y = 2i sinB => sinB = 1/2i(y – 1/y)
2cos(A + B) = 2(cosA cosB – sinA sinB)
= 2[1/2(x + 1/x) * 1/2(y + 1/y) – 1/2i(x – 1/x) * 1/2i(y – 1/y)]
= 2[1/4(xy + x/y + y/x + 1/xy + xy -x/y – y/x + 1/xy)]
 = xy + 1/xy

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