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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Trigonometric Equations

NTA Abhyas 2020 · Mathematics · Trigonometric Equations

The number of values of x lying in the interval (- 2π, 2π)...

The number of values of xx lying in the interval (−2π,2π)\left(- 2\pi, 2\pi\right) satisfying the equation 1+cos⁡10xcos6x=2cos⁡28x+sin⁡28x1 + \cos 10xcos 6x = 2\cos^{2}8x + \sin^{2}8x is equal to

Correct answer

7

Step-by-step solution

We have
1+cos⁡10xcos6x=2cos⁡28x+sin⁡28x1 + \cos 10xcos 6x = 2\cos^{2}8x + \sin^{2}8x
=cos⁡28x+1= \cos^{2}8x + 1
So, cos⁡10xcos6x=cos⁡28x=1+cos⁡16x2\cos 10xcos 6x = \cos^{2}8x = \dfrac{1 + \cos 16x}{2}
⇒cos⁡16x+cos⁡4x=1+cos⁡16x⇒ \cos 16x + \cos 4x = 1 + \cos 16x
⇒cos⁡4x=1⇒ \cos 4x = 1
⇒4x=2nπ,n∈Z⇒ 4x = 2n\pi, n \in Z
⇒x=nπ2,n∈Z⇒ x = \dfrac{n\pi}{2}, n \in Z
in \left(- 2\pi, 2\pi\right), x = \dfrac{\pm 3\pi}{2}, \pm \pi, \dfrac{\pm \pi}{2}, 0<br/>So,thenumberofvaluesof<br />So, the number of values ofxisis7 .
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