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JEE MATHEMATICS MCQ: TRIGONOMETRIC EQUATIONS

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If cos⁑A=mcos⁑B\cos A=m \cos B and cot⁑(A+B2)=Ξ»/tan⁑(Bβˆ’A2)\cot \left(\frac{A+B}{2}\right)=\lambda/\tan \left(\frac{B-A}{2}\right), then Ξ»\lambda is equal to
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Correct option: C
m+1mβˆ’1\frac{m+1}{m-1}
Option C (m+1mβˆ’1\frac{{m+1}}{{m-1}}) is correct because it's derived by applying the componendo and dividendo rule and trigonometric identities to the given equation.

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The number of solutions of the equation sin⁑  xβ€‰β€‰β€…β€Šcos  3x=sin⁑  3x  cos⁑  5x\sin \,\,x\,\,\;cos\,\,3x = \sin \,\,3x\,\,\cos \,\,5x in [0,β€…β€Šβ€‰Ο€2]\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⁑2βˆ’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 ΞΈβ†’Ο€2\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,Ο€4]x \in\left[0, \frac{\pi}{4}\right], then tan⁑x2\tan \frac{x}{2} is equal to The value of tan⁑π5+2tan⁑2Ο€5+4cot⁑4Ο€5\tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5} is If 0<x<Ο€0 < 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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