Which of the following inverse-trigonometric identities is/are correct? (i) sin⁻¹(3/5) + sin⁻¹(8/17) = cos⁻¹(36/85) (ii) tan⁻¹(3/4) + tan⁻¹(3/5) - tan⁻¹(8/19) = pi/4 (iii) tan⁻¹(2/11) + tan⁻¹(7/24) = tan⁻¹(1/2)

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All of (i), (ii), (iii). (i) sin⁻¹(3/5)+sin⁻¹(8/17): let A=sin⁻¹(3/5) (cosA=4/5), B=sin⁻¹(8/17) (cosB=15/17). cos(A+B)=cosA cosB - sinA sinB = (4/5)(15/17) - (3/5)(8/17) = (60-24)/85 = 36/85, so A+B = cos⁻¹(36/85). True. (ii) tan⁻¹(3/4)+tan⁻¹(3/5) = tan⁻¹((3/4+3/5)/(1-9/20)) = tan⁻¹((27/20)/(11/20)) = tan⁻¹(27/11). Then tan⁻¹(27/11)-tan⁻¹(8/19) = tan⁻¹((27/11-8/19)/(1+(27/11)(8/19))) = tan⁻¹((513-88)/(209+216)) = tan⁻¹(425/425) = tan⁻¹(1) = pi/4. True. (iii) tan⁻¹(2/11)+tan⁻¹(7/24) = tan⁻¹((2/11+7/24)/(1-(2/11)(7/24))) = tan⁻¹((48+77)/(264-14)) = tan⁻¹(125/250) = tan⁻¹(1/2). True. All three hold.

Which of the following inverse-trigonometric identities is/are correct? (i) sin⁻¹(3/5) + sin⁻¹(8/17) = cos⁻¹(36/85) (ii) tan⁻¹(3/4) + tan⁻¹(3/5) - tan⁻¹(8/19) = pi/4 (iii) tan⁻¹(2/11) + tan⁻¹(7/24) = tan⁻¹(1/2)

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