Question

sinA-sin 5A+ sin 9A - sin 13A÷
cosA - cos 5A - cos 9A+ cos 13A
= cot 4A​

Answer 1

Let the present ages of A and B be x and y respectively.

According to first condⁿ

$\frac{x}{y} = \frac{2}{3} \\ x = \frac{2}{3} y \\ y = \frac{3x}{2} \: \: \: \: \: \: (i)$

(always try to get denominators in factors of 2, 5 or 10)

now,

According to second condⁿ

$\frac{x + 5}{y + 5} = \frac{3}{4} \\ 4x + 20 = 3y + 15 \\ 20 - 15 = 3y - 4x \\ 3\frac({3x}{2}) - 4x = 5 \: \: \: \: \: [from (i)] \\ \frac{9x}{2} - 4x = 5 \\ multiplying throughout by 2, \\ 9x - 8x = 10 \\ x = 10 \\ \\ subsituting value of x in (i) \\ y = \frac{3(10)}{2} = \frac{30}{2} \\ y = 15$

Hence, present ages of A and B are 10 and 15 years respectively.