Last updated at May 12, 2021 by Teachoo
Transcript
Misc 3 Prove 2 sin-1 3/5 = tan-1 24/7 We need to convert LHS in form tan-1 Converting sin-1 (π/π) to tan-1 Let x = sin-1 (3/5) sin x = 3/5 Now, cos x = β(1βπ ππ2 π₯) = β(1 β (3/5)^2 ) = β(1 β 9/25) = β((25 β 9)/25) = β(16/25) = 4/5 Thus, tan x = sinβ‘π₯/cosβ‘π₯ tan x = (3/5)/(4/5) tan x = 3/4 x = tanβ1 π/π Solving L.H.S 2 sinβ1 π/π = 2x = 2 tan-1 (3/4) Using 2tan-1 x = tan-1 (ππ/(π β ππ)) = tan-1 (2(3/4)/(1 β (3/4)2)) = tan-1 ((3/2)/(1 β 9/16)) = tan-1 ((3/2)/( (16 β 9)/16)) = tan-1 ((3/2)/( 7/16)) = tan-1 (3/2Γ16/7) = tan-1 (ππ/π) = R.H.S. Hence L.H.S. = R.H.S Hence proved
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