Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2026
Exam
NDA
Year
2026
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2026 · Mathematics · Trigonometric Ratios & Identities

If 2 4β= 2α+ 2α , then which one of the following is a possible...

If 2sec⁡4β=tan⁡2α+cot⁡2α2\sec 4\beta=\tan 2\alpha+\cot 2\alpha, then which one of the following is a possible value of (α+β)(\alpha+\beta) ?

Options

  1. Aπ2\dfrac{\pi}{2}
  2. Bπ4\dfrac{\pi}{4}
  3. Cπ6\dfrac{\pi}{6}
  4. Dπ8\dfrac{\pi}{8}

Correct answer

D. π8\dfrac{\pi}{8}

Step-by-step solution

The given equation is 2sec⁡4β=tan⁡2α+cot⁡2α2\sec 4\beta = \tan 2\alpha + \cot 2\alpha

We know that tan⁡θ+cot⁡θ=sin⁡θcos⁡θ+cos⁡θsin⁡θ=1sin⁡θcos⁡θ=2sin⁡2θ\tan \theta + \cot \theta = \dfrac{\sin \theta}{\cos \theta} + \dfrac{\cos \theta}{\sin \theta} = \dfrac{1}{\sin \theta \cos \theta} = \dfrac{2}{\sin 2\theta}

Substituting θ=2α\theta = 2\alpha, we get tan⁡2α+cot⁡2α=2sin⁡4α\tan 2\alpha + \cot 2\alpha = \dfrac{2}{\sin 4\alpha}

The equation becomes 2sec⁡4β=2sin⁡4α2\sec 4\beta = \dfrac{2}{\sin 4\alpha}

⇒1cos⁡4β=1sin⁡4α\Rightarrow \dfrac{1}{\cos 4\beta} = \dfrac{1}{\sin 4\alpha}

⇒cos⁡4β=sin⁡4α\Rightarrow \cos 4\beta = \sin 4\alpha

⇒cos⁡4β=cos⁡(π2−4α)\Rightarrow \cos 4\beta = \cos\left(\dfrac{\pi}{2} - 4\alpha\right)

The general solution is 4β=2nπ±(π2−4α)4\beta = 2n\pi \pm \left(\dfrac{\pi}{2} - 4\alpha\right) where nn is an integer.

Taking the positive sign, we get:

4β=2nπ+π2−4α4\beta = 2n\pi + \dfrac{\pi}{2} - 4\alpha

⇒4(α+β)=2nπ+π2\Rightarrow 4(\alpha + \beta) = 2n\pi + \dfrac{\pi}{2}

⇒α+β=nπ2+π8\Rightarrow \alpha + \beta = \dfrac{n\pi}{2} + \dfrac{\pi}{8}

For n=0n = 0, a possible value is α+β=π8\alpha + \beta = \dfrac{\pi}{8}

Answer: π8\dfrac{\pi}{8}
Practice Trigonometric Ratios & Identities on Quantrex Academy →

More from Trigonometric Ratios & Identities

If θ lies in the fourth quadrant and 3 θ+4=0 , then what is the value of 2θ+ 2θ ? NDA 2026If α+ β=0= α+ β , α≠β then what is a value of 2α+ 2β+2 (α+β) ? NDA 2026If x= θ- θ and y= θ+ θ , then which one of the following is correct ? NDA 2026If θ=(1)/(3) , then what is the value of ((θ)/(2)) ((3θ)/(2)) ? NDA 2026If A and B are acute angles such that 2A+2B=π , then what is the maximum value of A· B ? NDA 2026x+ 3 x is maximum when x is equal to NDA 2026In a triangle ABC , A= B+ C then what is ((B)/(2))+ ((B)/(2))equal to ? NDA 2026If α and β are complementary angles such that α-β=(π)/(6) and m β=n α , then what is… NDA 2026For the following two (02) items: Let A + B = p and A + B = q . What is (p^2 - q^2)/(p^2… NDA 2025For the following two (02) items: Let p = cosec\, 20° and q = cosec\, 70° . What is (p^2… NDA 2025For the following two (02) items: Let A + B = p and A + B = q . What is (p)/(q) equal to? NDA 2025For the following two (02) items: Let p = cosec\, 20° and q = cosec\, 70° . What is ( 3… NDA 2025Consider the following for the three (03) items that follow: Let p = 35° , q = 25° and r… NDA 2025Consider the following for the three (03) items that follow: Let p = 35° , q = 25° and r… NDA 2025Consider the following for the three (03) items that follow: Let p = 35° , q = 25° and r… NDA 2025Consider the following for the two (02) items that follow: Let p = | α - (α - 90°)| .… NDA 2025 Full Trigonometric Ratios & Identities list Trigonometric Ratios & Identities JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs