Suppose \(0^\circ<\theta<90^\circ\) and the difference \(\mathrm{cosec}\,\theta-\cot\theta\) works out to \(\displaystyle\frac{1}{3}\). Using this, determine \(\cos\theta\).
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{4}{5}\)
The identity \(\mathrm{cosec}^2\theta-\cot^2\theta=1\) factorises as \((\mathrm{cosec}\,\theta-\cot\theta)(\mathrm{cosec}\,\theta+\cot\theta)=1\). So the sum is the reciprocal of the given difference, and the two equations give \(\mathrm{cosec}\,\theta\) and \(\cot\theta\).
\(\mathrm{cosec}\,\theta+\cot\theta=3\)
Adding the two equations: \(2\,\mathrm{cosec}\,\theta=3+\displaystyle\frac{1}{3}=\frac{10}{3}\), so \(\mathrm{cosec}\,\theta=\displaystyle\frac{5}{3}\)
Subtracting: \(2\cot\theta=3-\displaystyle\frac{1}{3}=\frac{8}{3}\), so \(\cot\theta=\displaystyle\frac{4}{3}\)
\(\cos\theta=\cot\theta\times\sin\theta=\displaystyle\frac{4}{3}\times\frac{3}{5}=\frac{4}{5}\)