Assertion Consider =\sin ^{-1}\left(\sec \left(\tan ^{-1} \boldsymbol{x}\right)+\right. \) \right. \) Statement-1: Domain of \) is a singleton. Reason Statement-2: Range of the function \) is a singleton.

Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.

Maths · Inverse trigonometrical functions and their properties

Assertion Consider =\sin ^{-1}\left(\sec \left(\tan ^{-1} \boldsymbol{x}\right)+

Assertion Consider \( \boldsymbol{f}(\boldsymbol{x})=\sin ^{-1}\left(\sec \left(\tan ^{-1} \boldsymbol{x}\right)+\right. \) \( \cos ^{-1}\left(\operatorname{cosec}\left(\cot ^{-1} x\right)\right. \) Statement-1: Domain of \( f(x) \) is a singleton. Reason Statement-2: Range of the function \( \boldsymbol{f}(\boldsymbol{x}) \) is a singleton.

A. Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1.
B. Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1.
C. Statement- lis true, Statement-2 is false.
D. Statement-1 is false, Statement-2 is true

Step-by-step solution

The function simplifies to f(x) = sin^{-1}(√(1+x^2)) + cos^{-1}(√(1+x^2)). For both inverse trigonometric functions to be defined, √(1+x^2) must be in [-1,1], which forces x=0. Hence domain is {0} (singleton). At x=0, f(0)=π/2, so range is {π/2} (singleton). Statement-2 is true but does not explain why domain is a singleton; rather, the singleton domain implies singleton range.

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Step-by-step solution

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