Inverse Trigonometric Functions

The number of solutions of \({\text{tan}}^{−1}4x+{\text{tan}}^{−1}6x=\frac{\pi }{6}\), where \(−\frac{1}{2\sqrt{6}}

[JEE Main 2026, 22 Jan (Shift 1)]

If \(f(x)=16\left(\left(\sec ^{-1} x\right)^2+\left(\operatorname{cosec}^{-1} x\right)^2\right)\) then the sum of max. and min. value of \(f(x)\) is (22 Jan, Shift I, Memory Based)

If \(\mathrm{y}={\sin }^{-1}x,-1\leq x\leq 0\), then the range of y is:

If \(\mathrm{y}={\sin }^{-1}x,-1\leq x\leq 0\), then the range of \(y\) is:

(ii) The principal value of \({\sin }^{-1}\left(-\frac{1}{2}\right)\) is -

The principal value of \({\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)\)is

Assertion (A) : Domain of \(\mathrm{y}={\cos }^{-1}(\mathrm{x})\) is \([-1,1]\).
Reason (R) : The range of the principal value branch of \(\mathrm{y}={\cos }^{-1}(x)\) is \([0,\pi ]-\left\{\frac{\pi }{2}\right\}\).

Direction : Two statements are given one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the following options :
Assertion (A) : Domain of \(\mathrm{y}={\cos }^{-1}(\mathrm{x})\) is \([-1,1]\).
Reason (R) : The range of the principal value branch of \(\mathrm{y}={\cos }^{-1}(x)\) is \([0,\pi ]-\left\{\frac{\pi }{2}\right\}\).

\({\tan }^{-1}\sqrt{3}-{\sec }^{-1}(-2)\) is equal to