Question

The value of \( \dfrac{\cos 37^\circ}{\sin 53^\circ} + \dfrac{\cot 34^\circ}{\tan 56^\circ} \) is:

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(-1\)

Hint:

Co-function pairs: \(\sin(90^\circ-\theta)=\cos\theta\), \(\tan(90^\circ-\theta)=\cot\theta\), \(\sec(90^\circ-\theta)=\csc\theta\). They make many evaluations trivial.

Answer 1

The Correct Option is C

Step 1: Use co-function identities.
Since \(53^\circ = 90^\circ - 37^\circ\), we have \(\sin 53^\circ = \cos 37^\circ\).
Thus, \(\dfrac{\cos 37^\circ}{\sin 53^\circ} = \dfrac{\cos 37^\circ}{\cos 37^\circ} = 1.\)
Step 2: Simplify the second term.
Because \(56^\circ = 90^\circ - 34^\circ\), \(\tan 56^\circ = \cot 34^\circ\).
Hence, \(\dfrac{\cot 34^\circ}{\tan 56^\circ} = \dfrac{\cot 34^\circ}{\cot 34^\circ} = 1.\)
Step 3: Add the results.
Total \(= 1 + 1 = 2.\)