Question

If y = √(tan √x), prove that √x (dy/dx) = (1 + y⁴)/(4y).

If a function f : X → Y defined as f(x) = y is one-one and onto, then we can define a unique function g : Y → X such that g(y) = x, where x ∈ X and y = f(x), y ∈ Y. Function g is called the inverse of function f. The domain of sine function is R and function sine : R → R is neither one-one nor onto. The following graph shows the sine function. Let sine function be defined from set A to [– 1, 1] such that inverse of sine function exists, i.e., sin⁻¹ x is defined from [– 1, 1] to A. On the basis of the above information, answer the following questions :

Find the value of tan⁻¹(–1/√3) + cot⁻¹(1/√3) + tan⁻¹[sin(–π/2)].

Find the domain of the function f(x) = sin⁻¹(x² – 4). Also, find its range.