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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 solutions of the equation ta n ^ 2 x - se c ^ 10 x +...

The number of solutions of the equation tan2x−sec10x+1=0ta{n}^{2}x - se{c}^{10}x + 1 = 0 for $x \in \left(0,20\right) is equal to

Correct answer

6

Step-by-step solution

sec2x−sec10x=0se{c}^{2}x - se{c}^{10}x = 0
sec2x(1−sec8x)=0se{c}^{2}x\left(1 - se{c}^{8}x\right) = 0
1−sec8x=01 - se{c}^{8}x = 0 as sec2x≥1se{c}^{2}x \ge 1
⇒sec8x=1⇒cos8x=1⇒cos⁡x=±1⇒ se{c}^{8}x = 1 ⇒ co{s}^{8}x = 1 ⇒ \cos x = \pm 1
⇒x=nπ,∀n∈I⇒ x = n\pi, \forall n \in Ι
⇒x={π,2π,3π,4π,5π,6π}∈(0,20)⇒ x = \left\{\pi, 2\pi, 3\pi, 4\pi, 5\pi, 6\pi\right\} \in \left(0,20\right)
So, the required number of roots =6= 6
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