Question

Cosec x - cot x = 3/2

Answer 1

$\huge\mathcal{Heyy!!!}\\Given\\csc x\:-\:cot x = \frac{3}{2}====>\:(EQ\:01)\\we\:know\:that\\csc^2x\:-\:cot^2x=1\\(csc x+cot x)(csc x\:-\:cot x)=1\\\bold{a^2-b^2=(a+b)(a-b)}\\csc x +cot x=\frac{1}{csc x\:-\:cot x}\\csc x +cot x=\frac{2}{3}====>\:(EQ\:02)\\add\:eq's\:01\:and\:02\\we\:get\:\\2.csc x=\frac{3}{2}+\frac{2}{3}\\2.csc x=\frac{13}{6}\\csc x = \frac{13}{12}\\\frac{1}{sin x }=\frac{13}{12}\\sin x =\frac{12}{13}\\we\:know\:that\:\\cos x =\sqrt{1-sin^2x}\\cos x =\sqrt{1-\left(\frac{12}{13}\right)^2}</p><p>\\</p><p>=\sqrt{1-\frac{144}{169}}\\=\sqrt{\frac{169-144}{169}}</p><p>\\</p><p>=\sqrt\frac{25}{169}</p><p>\\</p><p>\huge\mathcal{cos\:x= \frac{5}{13}}$

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