Evaluate the following: (i) cot(tan⁻¹ a + cot⁻¹ a); (ii) sin(sin⁻¹ x + cos⁻¹ x) for |x|

Question

Evaluate the following: (i) cot(tan⁻¹ a + cot⁻¹ a); (ii) sin(sin⁻¹ x + cos⁻¹ x) for |x| <= 1.

Accepted Answer

(i) 0; (ii) 1. For all real a, tan⁻¹ a + cot⁻¹ a = pi/2, so cot(pi/2) = 0. For all |x| <= 1, sin⁻¹ x + cos⁻¹ x = pi/2, so sin(pi/2) = 1.

Evaluate the following: (i) cot(tan⁻¹ a + cot⁻¹ a); (ii) sin(sin⁻¹ x + cos⁻¹ x) for |x| <= 1.