Considering only the principal values of the inverse trigonometric functions, the value of ( ⁻¹ ( 3 5 )-2 ⁻¹ ( 2 5 ) ) is

Step-by-step solution

Let E = ( ⁻¹ ( 3 5 ) - 2 ⁻¹ ( 2 5 ) ) . Evaluate the inverse trigonometric expressions: Let = ⁻¹ ( 3 5 ) , so = 3 5 and = 3 4 . Let = ⁻¹ ( 2 5 ) , so = 2 5 and = 1 2 . Compute the double angle: Using (2 ) = 2 1 - ^2 , we obtain (2 ) = 2 1 2 1 - ( 1 2 )^2 = 4 3 . Apply the tangent subtraction formula: E = ( - 2 ) = - (2 ) 1 + (2 ) = 3 4 - 4 3 1 + 3 4 4 3 = - 7 12 2 = - 7 24 . Final result: - 7 24