CBSE Class 12 Sample Paper for 2018 Boards

Class 12
Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

### Prove that if 1/2 β€ x β€ 1, then cos β1 x + cos β1 [x/2+β(3 - 3x 2 )/2] = Ο/3

This is a question of CBSE Sample Paper - Class 12 - 2017/18.

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### Transcript

Question 7 Prove that if 1/2 β€ x β€ 1 then cosβ1 x + cosβ1 [π₯/2+β(3 β 3π₯^2 )/2] = π/3 Solving LHS cosβ1 x + cosβ1 [π₯/2+β(3 β 3π₯^2 )/2] = cosβ1 x + cosβ1 [π₯/2+β(3(1 β π₯^2))/2] Let x = cos π i.e. cosβ1 x = π Theory Since it is β(1βπ₯^2 ) we can use x = sin π or x = cos π Since it is given cos-1 x, we use x = cos π = π + cosβ1 [cosβ‘π/2+β(3(1βcos^2β‘γπ)γ )/2] Using sin ΞΈ = 1 β cos2 ΞΈ = π + cosβ1 [cosβ‘π/2+β(3 sin^2β‘π )/2] = π + cosβ1 [cosβ‘π/2+γβ3 sinγβ‘π/2] = π + cosβ1 [1/2 Γcosβ‘π+β3/2 Γsinβ‘π ] Theory We convert 1/2 Γcosβ‘π+β3/2 Γsinβ‘π into cos (x β y) Since cos (x β y) = cos x cos y + sin x sin y We use x = 60Β° and y = π = π + cosβ1 [cosβ‘γ60Β°γ Γcosβ‘π+sinβ‘γ60Β°γΓsinβ‘π ] = π + cosβ1 [cosβ‘γπ/3γ Γcosβ‘π+sinβ‘γπ/3γΓsinβ‘π ] Using cos (x β y) = cos x cos y + sin x sin y = π + cosβ1 [cosβ‘(π/3βπ) ] = π + cosβ1 [πππ (π/3βπ)] = π + π/3 β π = π/3 = RHS Hence proved