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JEE MATHEMATICS MCQ: TRIGONOMETRIC EQUATIONS

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The value of the expression sin⁡3x1+cos⁡x+cos⁡3x1−sin⁡x\frac{\sin ^3 x}{1+\cos x}+\frac{\cos ^3 x}{1-\sin x} is
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Correct option: A
2cos⁡[π4−x]\sqrt{2} \cos \left[\frac{\pi}{4}-x\right]
Option A (2cos⁡[π4−x]\sqrt{2} \cos \left[\frac{\pi}{4}-x\right]) is correct because the expression simplifies to sin⁡x+cos⁡x\sin x + \cos x, which is equivalent to 2cos⁡[π4−x]\sqrt{2} \cos \left[\frac{\pi}{4}-x\right].

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The number of solutions of the equation sin⁡  x    cos  3x=sin⁡  3x  cos⁡  5x\sin \,\,x\,\,\;cos\,\,3x = \sin \,\,3x\,\,\cos \,\,5x in [0,   π2]\left[ {0,\;\,\frac{\pi }{2}} \right] is The number of ordered pairs (x, y)\left( {x,\,y} \right) where x,  y∈[0,10]x,\,\,y \in \left[ {0,10} \right] satisfying (sin⁡2−sin⁡  x+12)2sec⁡2y  ⋅  ≤1\left( {\sqrt {{{\sin }^2} - \sin \,\,x + \frac{1}{2}} } \right){2^{{{\sec }^2}y}}\,\, \cdot \,\, \le 1 is Statement I: If slope of the tangent line is zero, then tangent line is perpendicular to the X\mathrm{X}-axis.

Statement II: If θ→π2\theta \rightarrow \frac{\pi}{2}, then tangent line is parallel to the Y-axis.
 If sin⁡x+cos⁡x=72\sin x+\cos x=\frac{\sqrt{7}}{2}, where x∈[0,π4]x \in\left[0, \frac{\pi}{4}\right], then tan⁡x2\tan \frac{x}{2} is equal to The value of tan⁡π5+2tan⁡2π5+4cot⁡4π5\tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5} is If 0<x<π0 < x < \pi and cos⁡x+sin⁡x=12\cos x+\sin x=\frac{1}{2}, then tan⁡x\tan x is equal to
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