If \(3\sin\theta-4\cos\theta=0\), then value of \(\tan\theta\cdot\mathrm{cosec}\,\theta\) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\frac{5}{3}\)
Since \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) and \(\mathrm{cosec}\,\theta=\frac{1}{\sin\theta}\), the sines cancel, and \(\tan\theta\cdot\mathrm{cosec}\,\theta=\frac{1}{\cos\theta}=\sec\theta\). So only \(\sec\theta\) is needed.
\(3\sin\theta=4\cos\theta\), so \(\tan\theta=\frac{4}{3}\)
In a right triangle with opposite side 4 and adjacent side 3, the hypotenuse is \(\sqrt{3^{2}+4^{2}}=5\)
\(\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{5}{3}\)
