Question

If 2 cos q = x + $\frac{1}{x}$and 2 cos f = y +$\frac{1}{y}$, then $\frac{x}{y}$+ $\frac{y}{x}$equals

• A. 2 cos (q –f)
• B. 2 cos (q + f)
• C. 2 sin (q – f)
• D. 2 sin (q + f)

Answer 1

Answer: (A)

2 cos (q –f)

Answer 2

Sol. 2 cos q = x + $\frac{1}{x}$

Ž x² –(2 cos q) x + 1 = 0

Ž x = cos q ± i sin q

Similarly, y = cos f ± i sin f

taking x = cos q + i sin q and y = cos f + i sin f

$\frac{x}{y}$+$\frac{y}{x}$= $\frac{\cos\theta + i\sin\theta}{\cos\varphi + i\sin\varphi}$+ $\frac{\cos\varphi + i\sin\varphi}{\cos\theta + i\sin\theta}$

= {cos (q – f) + i sin (q – f)} + cos (f – q) + i sin (f – q)

= cos (q – f) + i sin(q – f) + cos (q – f) – i sin (q – f)

= 2 cos (q – f)