Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 2005
Exam
JEE Main
Year
2005
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

JEE Main 2005 · Mathematics · Inverse Trigonometric Functions

If ^ -1 x- ^ -1 (y)/(2)=α , then 4 x^2-4 x y α+y^2 is equal to

If cos⁡−1x−cos⁡−1y2=α\cos ^{-1} x-\cos ^{-1} \frac{y}{2}=\alpha, then 4x2−4xycos⁡α+y24 x^2-4 x y \cos \alpha+y^2 is equal to

Options

  1. A2sin⁡2α2 \sin 2 \alpha
  2. B4
  3. C4sin⁡2α4 \sin ^2 \alpha
  4. D−4sin⁡2α-4 \sin ^2 \alpha

Correct answer

C. 4sin⁡2α4 \sin ^2 \alpha

Step-by-step solution

cos⁡−1x−cos⁡−1y2=αcos⁡−1(xy2+(−x2)(1−y24))=αcos⁡−1(xy+4−y2−4x2+x2y22)=α⇒4−y2−4x2+x2y2=4cos⁡2α+x2y2−4xycos⁡α⇒4x2+y2−4xycos⁡α=4sin⁡2α\begin{aligned}& \cos ^{-1} x-\cos ^{-1} \frac{y}{2}=\alpha\\ & \cos ^{-1}\left(\frac{x y}{2}+\sqrt{\left(-x^2\right)\left(1-\frac{y^2}{4}\right)}\right)=\alpha\\ & \cos ^{-1}\left(\frac{x y+\sqrt{4-y^2-4 x^2+x^2 y^2}}{2}\right)=\alpha\\ & \Rightarrow 4-y²-4 x²+x² y²=4 \cos ^2 \alpha+x^2 y²-4 x y \cos \alpha\\ & \Rightarrow 4 x²+y²-4 x y \cos \alpha=4 \sin ^2 \alpha \end{aligned}
Practice Inverse Trigonometric Functions on Quantrex Academy →

More from Inverse Trigonometric Functions

Considering only the principal values of the inverse trigonometric functions, the value… JEE Advanced 2026Let α = 3 ^ -1 ((6)/(11)) and β = 3 ^ -1 ((4)/(9)) , where inverse trigonometric… JEE Main 2026If ( ^ -1 (x 2 )) = ( ^ -1 1-x^2 ) , x ∈ (0,1) , then the value of x is : JEE Main 2026Let [·] denote the greatest integer function. If the domain of the function f(x) = ^ -1… JEE Main 2026Let 0 < α < 1 , β = (1)/(3α) and ^ -1 (1-α) + ^ -1 (1-β) = (π)/(4) . Then 6(α + β) is… JEE Main 2026If (π)/(4) + ∑_ p=1 ^ 11 ^ -1 ( 2^ p-1 1 + 2^ 2p-1 ) = α , then α is equal to __________. JEE Main 2026If y = ^ -1 ((3 x - 4 x)/(4 x + 3 x)) + 2 ^ -1 ( x 1+ 1-x^2 ) , then (dy)/(dx) at x = 3 2… JEE Main 2026Considering the principal values of inverse trigonometric functions, the value of the… JEE Main 2026If k= ((π)/(4)+(1)/(2) ^ -1 ((2)/(3)))+ ((1)/(2) ^ -1 ((2)/(3))) , then the number of… JEE Main 2026If the domain of the function f(x)= ^ -1 ( 1 x^ 2 -2 x-2 ) , is (-∞, α] [β, γ] [δ, ∞)… JEE Main 2026The number of solutions of ^ -1 4x + ^ -1 6x = (π)/(6) , where - 1 2 6 < x < 1 2 6 , is… JEE Main 2026Let the maximum value of ( ^ -1 x)^ 2 +( ^ -1 x)^ 2 for x ∈[- 3 2 , 1 2 ] be m n π^ 2… JEE Main 2026If the domain of the function f(x)= ^ -1 ((2 x-5)/(11-3 x))+ ^ -1 (2 x^ 2 -3 x+1) is the… JEE Main 2026The total number of real solutions of the equation θ= ^ -1 (2 θ)-(1)/(2) ^ -1 ((6 θ)/(9+… JEE Advanced 2025The value of ^ -1 ( 1+ ^2(2) -1 (2) )- ^ -1 ( 1+ ^2((1)/(2)) +1 ((1)/(2)) ) is equal to JEE Main 2025The sum of the infinite series ^ -1 ((7)/(4))+ ^ -1 ((19)/(4))+ ^ -1 ((39)/(4))+ ^ -1… JEE Main 2025 Full Inverse Trigonometric Functions list Inverse Trigonometric Functions JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs