The CAT Online Free Test Prep MCQs Trigonometry - Fundamentals and Basic Identities

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In this problem, we can use the identities $\mathrm{cot}\theta =\frac{\mathrm{cos}\theta}{\mathrm{sin}\theta}$ and $\mathrm{csc}\theta =\frac{1}{\mathrm{sin}\theta}$

$(1-{\mathrm{cos}}^{2}\theta )\cdot \mathrm{cot}\theta \cdot {\mathrm{csc}}^{2}\theta =(1-{\mathrm{cos}}^{2}\theta )\cdot \frac{\mathrm{cos}\theta}{\mathrm{sin}\theta}\cdot (\frac{1}{\mathrm{sin}\theta}{)}^{2}$

$=(1-{\mathrm{cos}}^{2}\theta )\cdot \frac{\mathrm{cos}\theta}{\mathrm{sin}\theta}\cdot \frac{1}{{\mathrm{sin}}^{2}\theta}$

This doesn't seem to help a whole lot. However, we should recognize that (1−cos^{2}θ)=sin^{2}θ because of the Pythagorean identity sin^{2}θ+cos^{2}θ = 1 .

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