If \(\rm \frac{cos(x+y)}{cos(x-y)}=\frac{a+b}{a-b}\), What is tan(x)

1.	If \(\rm \frac{cos(x+y)}{cos(x-y)}=\frac{a+b}{a-b}\), What is tan(x) tan(y) equal to?1. b/a2. a/b3. -b/a4. 2b/a
Answer» Correct Answer - Option 3 : -b/a Concept: \(\rm cosA + cosB = 2cos(\frac{A+B}{2})cos(\frac{A-B}{2})\) \(\rm cosA - cosB = -2sin(\frac{A+B}{2})sin(\frac{A-B}{2})\) Calculation: Here, \(\rm \frac{\cos(x+y)}{\cos(x-y)}=\frac{a+b}{a-b}\) Applying componendo and dividendo, we get \(\rm \frac{cos(x+y)+cos(x-y)}{cos(x+y)-cos(x-y)}=\frac{a+b + a-b}{a+b-a + b}\) \(⇒ \rm \frac{2cos(\frac{x+y+x-y}{2})cos(\frac{x+y-x+y}{2})}{-2sin(\frac{x+y+x-y}{2})sin(\frac{x+y-x+y}{2})}=\frac{2a}{2b}\) ⇒ cot(x) cot(y) = -a/b ⇒ tan(x) tan(y) = -b/a Hence, option (3) is correct.

If \(\rm \frac{cos(x+y)}{cos(x-y)}=\frac{a+b}{a-b}\), What is tan(x) tan(y) equal to?1. b/a2. a/b3. -b/a4. 2b/a

Answer» Correct Answer - Option 3 : -b/a

Concept:

\(\rm cosA + cosB = 2cos(\frac{A+B}{2})cos(\frac{A-B}{2})\)

\(\rm cosA - cosB = -2sin(\frac{A+B}{2})sin(\frac{A-B}{2})\)

Calculation:

Here, \(\rm \frac{\cos(x+y)}{\cos(x-y)}=\frac{a+b}{a-b}\)

Applying componendo and dividendo, we get

\(\rm \frac{cos(x+y)+cos(x-y)}{cos(x+y)-cos(x-y)}=\frac{a+b + a-b}{a+b-a + b}\)

\(⇒ \rm \frac{2cos(\frac{x+y+x-y}{2})cos(\frac{x+y-x+y}{2})}{-2sin(\frac{x+y+x-y}{2})sin(\frac{x+y-x+y}{2})}=\frac{2a}{2b}\)

⇒ cot(x) cot(y) = -a/b

⇒ tan(x) tan(y) = -b/a

Hence, option (3) is correct.