Find the value of cosec 40°÷ sec 50°
To find the value of \(\csc 40^\circ \div \sec 50^\circ\), we can start by expressing both as fractions using the reciprocal identities.
Recall that \(\csc \theta = \frac{1}{\sin \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\).
\(\csc 40^\circ = \frac{1}{\sin 40^\circ}\)
\(\sec 50^\circ = \frac{1}{\cos 50^\circ}\)
Now, we can substitute these into the expression:
\(\frac{1}{\sin 40^\circ} \div \frac{1}{\cos 50^\circ}\)
To divide fractions, we can multiply by the reciprocal of the divisor:
\(\frac{1}{\sin 40^\circ} \cdot \frac{\cos 50^\circ}{1}\)
Simplifying, we have:
\(\frac{\cos 50^\circ}{\sin 40^\circ}\)
Using the trigonometric identity \(\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\), we can rewrite this as:
\(\frac{\cos(90^\circ - 40^\circ)}{\sin 40^\circ}\)
\(\frac{\cos 50^\circ}{\sin 40^\circ}\)
Now, we can use the reciprocal identity for \(\cos \theta = \frac{1}{\sec \theta}\) to simplify further:
\(\frac{1}{\frac{\sin 40^\circ}{\cos 50^\circ}}\)
\(\frac{1}{\tan 40^\circ}\)
Therefore, the value of \(\csc 40^\circ \div \sec 50^\circ\) is \(\frac{1}{\tan 40^\circ}\) or equivalently \(\text{cot}(40^\circ)\), the cotangent of \(40^\circ\).