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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 :
The derivative of tan⁻¹(x²) w.r.t. x is :
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.
Express tan⁻¹(cos x/(1 – sin x)), where –π/2 < x < π/2 in the simplest form.
Find the principal value of tan⁻¹(1) + cos⁻¹(–1/2) + sin⁻¹(–1/√2).
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