prove that sin-1 (-x) =-sin-1 x

Question

just prove

Answer 1

Prove sin^-1 (-x) =-sin^-1 x

 

Let u = sin^-1 (- x)         and v = - sin ^-1 x

Then, sin u = - x           and -v = sin ^-1 x  ⇒ x = sin(-v) = - sin v

       or x = - sin u 

 

Comparing the two,   x = - sin u =  - sin v  ⇒ u = v

 

Therefore,   sin^-1 (-x) = -sin^-1 x

Answer 2

I think you meant                  sin^-1(-x) = -sin^-1(x)
 
 
Rewrite using positive exponents.
 
1 / sin(-x) = -1 / sin(x)
  
 
Rewrite the left side using addition/subtraction angle identity.
 
1 / sin(0 - x) = -1 / sin(x)
 
1 / (sin(0)cos(x) - cos(0)sin(x)) = -1 / sin(x)
 
1 / -sin(x) = -1 / sin(x) 
 
-1 / sin(x) = -1 / sin(x)