Question

plzz ans it with step by step explanation​

Answer 1

Given :-

◉ sin ɑ = sin β ...(1)

◉ cos ɑ = cos β ...(2)

From these two equations, we conclude

⇒ ɑ = β

From the first equation we have ,

⇒ sin ɑ = sin β

⇒ sin ɑ - sin β = 0

It is in the form of sin C - sin D,

We know,

⇒ sin C - sin D = 2 sin (C - D)/2 cos (C + D)/2

So, we have

⇒ 2 sin (ɑ - β)/2 cos (ɑ + β)/2 = 0

From this, we have either of these true

• sin (ɑ - β)/2 = 0
• cos (ɑ + β)/2 = 0

But, we are not sure which one is zero, So from the equation (2) , we have

⇒ cos ɑ = cos β

⇒ cos ɑ - cos β = 0

Similarly, It is in the form of cos C - cos D

We know,

⇒ cos C - cos D = - 2 sin (C + D)/2 sin(C - D)/2

So, we have

⇒ - 2 sin (ɑ + β)/2 sin(ɑ - β)/2 = 0

From this, we have either of these true

• sin (ɑ + β)/2 = 0
• sin (ɑ - β)/2 = 0

But we have, ɑ = β

Substituting,

⇒ sin (ɑ - β)/2 = 0

⇒ sin (ɑ - ɑ)/2 = 0

⇒ sin 0 = 0

⇒ 0 = 0

If you substitute ɑ = β in any other equations, you wont get zero. Hence, our answer is sin (ɑ - β)/2 = 0

∴ Option (3) is correct.