For x ∈ (0, (π)/(2)), if cos^ - 1 ((7)/(2)(1 + 2x) + (sin^ 2 x - 48cos^ 2 x) sin x)= x - ^ -1 (kcos x), then the value of k is equal to

Correct answer: C. 7 y = cos^ - 1 ((7)/(2) (1 + 2x) + (sin^ 2 x - 48cos^ 2 x) sin x) = cos^ - 1 ((7cos x)(cos x) + 1 - 49 ^ 2 x 1 - ^ 2 x ) = cos^ - 1 (cos x) - cos^ - 1 (7cos x) = x - cos^ - 1 (7cos x) Hence, the value ofk = 7

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Inverse Trigonometric Functions

NTA Abhyas 2020 · Mathematics · Inverse Trigonometric Functions

For x ∈ (0, (π)/(2)), if cos^ - 1 ((7)/(2)(1 + 2x) + (sin^ 2 x -...

For

x \in \left(0, \dfrac{\pi}{2}\right),

cos^{- 1}\left(\dfrac{7}{2}\left(1 + \cos 2x\right) + \sqrt{\left(sin^{2}x - 48cos^{2}x\right)}sin x\right)= x - \cos^{-1}\left(kcos x\right),

then the value of

k

is equal to

Options

A $1$
B $5$
C $7$
D $14$

Correct answer

C. $7$

Step-by-step solution

y = cos^{- 1}

\left(\dfrac{7}{2}

\left(1 + \cos 2x\right) + \sqrt{\left(sin^{2}x - 48cos^{2}x\right)}

sin x\right)

<br /><br />

= cos^{- 1} \left(\left(7cos x\right)\left(cos x\right)

+

\sqrt{1 - 49\cos^{2}x}\sqrt{1 - \cos^{2}x}\right)

<br /><br />

= cos^{- 1} \left(cos x\right)

- cos^{- 1} \left(7cos x\right)

= x - cos^{- 1} \left(7cos x\right)<br /><br />Hence,

the value ofk = 7

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 2026 Let α = 3 ^ -1 ((6)/(11)) and β = 3 ^ -1 ((4)/(9)) , where inverse trigonometric… JEE Main 2026 If ( ^ -1 (x 2 )) = ( ^ -1 1-x^2 ) , x ∈ (0,1) , then the value of x is : JEE Main 2026 Let [·] denote the greatest integer function. If the domain of the function f(x) = ^ -1… JEE Main 2026 Let 0 < α < 1 , β = (1)/(3α) and ^ -1 (1-α) + ^ -1 (1-β) = (π)/(4) . Then 6(α + β) is… JEE Main 2026 If (π)/(4) + ∑_ p=1 ^ 11 ^ -1 ( 2^ p-1 1 + 2^ 2p-1 ) = α , then α is equal to __________. JEE Main 2026 If 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 2026 Considering the principal values of inverse trigonometric functions, the value of the… JEE Main 2026 If k= ((π)/(4)+(1)/(2) ^ -1 ((2)/(3)))+ ((1)/(2) ^ -1 ((2)/(3))) , then the number of… JEE Main 2026 If the domain of the function f(x)= ^ -1 ( 1 x^ 2 -2 x-2 ) , is (-∞, α] [β, γ] [δ, ∞)… JEE Main 2026 The number of solutions of ^ -1 4x + ^ -1 6x = (π)/(6) , where - 1 2 6 < x < 1 2 6 , is… JEE Main 2026 Let the maximum value of ( ^ -1 x)^ 2 +( ^ -1 x)^ 2 for x ∈[- 3 2 , 1 2 ] be m n π^ 2… JEE Main 2026 If 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 2026 The total number of real solutions of the equation θ= ^ -1 (2 θ)-(1)/(2) ^ -1 ((6 θ)/(9+… JEE Advanced 2025 The value of ^ -1 ( 1+ ^2(2) -1 (2) )- ^ -1 ( 1+ ^2((1)/(2)) +1 ((1)/(2)) ) is equal to JEE Main 2025 The 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 PYQs JEE Mathematics PYQs All NTA Abhyas JEE Main PYQs