Considering only the principal values of the inverse trigonometric functions, the value of ⁻¹( (-11)) + 10 (2 ⁻¹ ( 1 2 ) ) + 10 (2 ⁻¹(2)) is

Step-by-step solution

For the first term, ⁻¹( (-11)) : The principal value branch of ⁻¹(x) is (0, ) . Since the cotangent function is periodic with period , (-11) = (-11 + 4 ) . Using 3.14 , we have 4 12.56 , so -11 + 4 1.56 , which lies in the interval (0, ) . Thus, ⁻¹( (-11)) = 4 - 11 . For the second term, 10 (2 ⁻¹ ( 1 2 ) ) : Since ⁻¹ ( 1 2 ) = 4 , we get: 10 (2 4 ) = 10 ( 2 ) = 10(1) = 10 . For the third term, 10 (2 ⁻¹(2)) : Let = ⁻¹(2) ( ) = 2 . Using the multiple angle formula (2 ) = 2 ( ) 1 + ^2( ) , we obtain: (2 ) = 2(2) 1 + 2