(i) Solve the inequality tan⁻¹ x cot⁻¹ x. (ii) Find the complete solution set of the inequality (cos⁻¹ x)² - (sin⁻¹ x)² 0. Choose the correct answer for part (ii).

Question

(i) Solve the inequality tan⁻¹ x > cot⁻¹ x. (ii) Find the complete solution set of the inequality (cos⁻¹ x)² - (sin⁻¹ x)² > 0. Choose the correct answer for part (ii).

Accepted Answer

(B) [-1, 1/sqrt(2)). (ii) Write (cos⁻¹ x)² - (sin⁻¹ x)² = (cos⁻¹ x - sin⁻¹ x)(cos⁻¹ x + sin⁻¹ x). Since cos⁻¹ x + sin⁻¹ x = pi/2 > 0 for all x in [-1, 1], the sign is determined by cos⁻¹ x - sin⁻¹ x > 0, i.e. cos⁻¹ x > sin⁻¹ x. Using cos⁻¹ x = pi/2 - sin⁻¹ x, this becomes pi/2 - sin⁻¹ x > sin⁻¹ x -> sin⁻¹ x < pi/4 -> x < sin(pi/4) = 1/sqrt(2). Combined with the domain x in [-1, 1], the solution is [-1, 1/sqrt(2)). For (i), tan⁻¹ x > cot⁻¹ x = pi/2 - tan⁻¹ x leads to tan⁻¹ x > pi/4, i.e. x > 1. So (ii) answer is (B).

(i) Solve the inequality tan⁻¹ x > cot⁻¹ x. (ii) Find the complete solution set of the inequality (cos⁻¹ x)² - (sin⁻¹ x)² > 0. Choose the correct answer for part (ii).

Solution

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