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

JEE Main · Mathematics · Inverse Trigonometric Functions

If ^ -1 (1)/(5)+(1)/(2) ^ -1 x+ ^ -1 (1)/(8)=(π)/(8) , then x^2=

If tan⁡−115+12sec⁡−1x+tan⁡−118=π8\tan ^{-1} \frac{1}{5}+\frac{1}{2} \sec ^{-1} x+\tan ^{-1} \frac{1}{8}=\frac{\pi}{8}, then x2=x^2=

Options

  1. A127\frac{12}{7}
  2. B5049\frac{50}{49}
  3. C1312\frac{13}{12}
  4. D12\frac{1}{2}

Correct answer

B. 5049\frac{50}{49}

Step-by-step solution

We have, tan⁡−115+12sec⁡−1x+tan⁡−118=π8⇒(tan⁡−115+tan⁡−118)+12sec⁡−1x=π8\begin{array}{r} \tan ^{-1} \frac{1}{5}+\frac{1}{2} \sec ^{-1} x+\tan ^{-1} \frac{1}{8}=\frac{\pi}{8} \\ \Rightarrow \quad\left(\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{8}\right)+\frac{1}{2} \sec ^{-1} x=\frac{\pi}{8} \end{array}\Rightarrow \quad \tan ^{-1}\left[\frac{\frac{1}{5}+\frac{1}{8}}{1-\frac{1}{5} \times \frac{1}{8}}\right]+\frac{1}{2} \sec ^{-1}x=x=\frac{\pi}{8}⇒ and tan⁡−11339+12sec⁡−1x=π8⇒ and 2tan⁡−11339+sec⁡−1x=π4⇒ and 2tan⁡−113+sec⁡−1x=π4⇒ and tan⁡−1231−19+sec⁡−1x=π4⇒ and tan⁡−134+sec⁡−1x=π4\begin{array}{ll}\Rightarrow \text{ and } \tan ^{-1} \frac{13}{39}+\frac{1}{2} \sec ^{-1}x=\frac{\pi}{8}\\ \Rightarrow \text{ and } 2 \tan ^{-1} \frac{13}{39}+\sec ^{-1}x=\frac{\pi}{4}\\ \Rightarrow \text{ and } 2 \tan ^{-1} \frac{1}{3}+\sec ^{-1}x=\frac{\pi}{4}\\ \Rightarrow \text{ and } \tan ^{-1} \frac{\frac{2}{3}}{1-\frac{1}{9}}+\sec ^{-1}x=\frac{\pi}{4}\\ \Rightarrow \text{ and } \tan ^{-1} \frac{3}{4}+\sec ^{-1}x=\frac{\pi}{4}\end{array}⇒tan⁡−134+tan⁡−1x2−1=π4⇒tan⁡−1[34+x2−11−34x2−1]=π4\begin{array}{rr}\Rightarrow \quad & \tan ^{-1} \frac{3}{4}+\tan ^{-1} \sqrt{x^2-1}=\frac{\pi}{4} \\ \Rightarrow \quad & \tan ^{-1}\left[\frac{\frac{3}{4}+\sqrt{x^2-1}}{1-\frac{3}{4} \sqrt{x^2-1}}\right]=\frac{\pi}{4}\end{array}⇒ and 3+4x2−14−3x2−1=1⇒ and 3+4x2−1=4−3x2−1⇒7x2−1=1⇒ and x2−1=17\Rightarrowx2−1=149\Rightarrowx2=5049\begin{array}{cc}\Rightarrow \text{ and } \frac{3+4 \sqrt{x^2-1}}{4-3 \sqrt{x^2-1}}=1 \\ \Rightarrow \text{ and } 3+4 \sqrt{x^2-1}=4-3 \sqrt{x^2-1} \Rightarrow7 \sqrt{x^2-1}=1 \\ \Rightarrow \text{ and } \sqrt{x^2-1}=\frac{1}{7} \Rightarrowx^2-1=\frac{1}{49} \Rightarrowx^2=\frac{50}{49}\end{array}
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 99 Percentile Qs Bank for JEE Main PYQs