If x ( , 3 2 ) satisfies the equation ⁻¹ ( 7 5 ) - ⁻¹ ( 3 5 ) = ⁻¹( x) , then the value of x is

Step-by-step solution

We first evaluate the left-hand side of the given equation. For the first term, the principal value branch of ⁻¹x is [0, ] . Since 7 5 [0, ] , we can write: ⁻¹ ( 7 5 ) = ⁻¹ ( (2 - 7 5 ) ) = ⁻¹ ( 3 5 ) Since 3 5 [0, ] , we have ⁻¹ ( 7 5 ) = 3 5 . For the second term, the principal value branch of ⁻¹x is [- 2 , 2 ] . Since 3 5 [- 2 , 2 ] , we can write: ⁻¹ ( 3 5 ) = ⁻¹ ( ( - 3 5 ) ) = ⁻¹ ( 2 5 ) Since 2 5 [- 2 , 2 ] , we have ⁻¹ ( 3 5 ) = 2 5 . Substituting these values into the equation, we get: 3 5 - 2 5 = ⁻¹( x) ⁻