Question

If `sin^-1 x + sin^-1 y+sin^-1 z+sin^-1 t=2pi` , then find the value of x² + y² + z² + t² 

Answer 1

We know that the maximum value of sin^-1 x, sin^-1 y, sin^-1 z and sin^-1 t is  `pi/2`
Now,

LHS = `sin^-1 x+sin^-1y+sin^-1z+sin^-1t`

`=pi/2+pi/2+pi/2+pi/2`

= 2 π = RHS

Now,
`sin^-1x=pi/2,sin^-1y=pi/2,sin^-1z=pi/2` and  `sin^-1t=pi/2`

⇒ `x = sin  x/2,y=sin  pi/2, z=sin  pi/2 and t=sin  pi/2`

⇒ x = 1, y = 1, z = 1, and t = 1

∴ x² + y² + z² + t² = 1 + 1 + 1 + 1 = 4