Question

If sin x + sin y = $\frac{1}{4}$ and cos x + cos y = $\frac{1}{3}$, then show that(i). $tan(\frac{x + y}{2}) = \frac{3}{4}$(ii). cot (x + y) = $\frac{7}{24}$

Answer 1

Answer:

Step-by-step explanation:

Given:

sin x + sin y = 1/4

 and cos x + cos y = 1/3

1.

sin x + sin y = 1/4

implies

2 sin((x+y)/2) cos((x-y)/2) = 1/4......(1)

cos x + cos y = 1/3

implies

2 cos((x+y)/2) cos((x-y)/2) = 1/3......(2)

Divide (1) by (2)

tan((x+y)/2) = (1/4)×(3/1)

 tan((x+y)/2) = 3/4

2.

Now

cot(x+y)

= cot(2((x+y)/2))

= cot(2t) where t = = (x+y)/2

= (1/tan2t) = 7/24

because

tan2t

= 2 tant /(1 - tan²t)

= 2(3/4) / (1-(9/16))

= (3/2)/(7/16)

= 48/14

= 24/7

Answer 2

Answer:

here's your answer

Step-by-step explanation:

refer to the picture