Prove that sin-1(3/5) - sin-1(8/17) = cos-1(84/85)

1.	Prove that sin-1(3/5) - sin-1(8/17) = cos-1(84/85)
Answer» Let sin^-1(3/5) = x and sin^-1(8/17) = y Therefore sinx = 3/5 and siny = 8/17 Now, cosx = √(1 - sin²x) = √(1 - (3/5)²) = √(1 - 9/25) = 4/5 and cosy = √(1 - sin²y) = √(1 - (8/17)²) = √(1 - 64/289) = 15/17 We have cos(x - y) = cosx cosy + sinx siny = 4/5 x 15/17 + 3/5 x 8/17 = 60/85 + 24/85 = 84/85 ⇒ x - y = cos^-1(84/85) ⇒ sin^-1(3/5) - sin^-1(8/17) = cos^-1(84/85)

Answer»

Let sin^-1(3/5) = x and sin^-1(8/17) = y

Therefore sinx = 3/5 and siny = 8/17

Now, cosx = √(1 - sin²x) = √(1 - (3/5)²) = √(1 - 9/25) = 4/5

and cosy = √(1 - sin²y) = √(1 - (8/17)²) = √(1 - 64/289) = 15/17

We have cos(x - y) = cosx cosy + sinx siny

= 4/5 x 15/17 + 3/5 x 8/17 = 60/85 + 24/85 = 84/85

⇒ x - y = cos^-1(84/85)

⇒ sin^-1(3/5) - sin^-1(8/17) = cos^-1(84/85)

Discussion