Question

Evaluate \( \csc(90^\circ - \theta) \cdot \cos(90^\circ - \theta) \):

• A. \( \sec \theta \)
• B. \( \tan \theta \)
• C. \( \sin \theta \)
• D. \( \cot \theta \)

Hint:

Using complementary angle identities: \[ \sin(90^\circ - \theta) = \cos \theta, \quad \cos(90^\circ - \theta) = \sin \theta. \] \[ \csc(90^\circ - \theta) = \sec \theta, \quad \sec(90^\circ - \theta) = \csc \theta. \]

Answer 1

The Correct Option is B

Using trigonometric identities:
\[ \csc(90^\circ - \theta) = \sec \theta, \quad \cos(90^\circ - \theta) = \sin \theta. \] \[ \csc(90^\circ - \theta) \cdot \cos(90^\circ - \theta) = \sec \theta \cdot \sin \theta. \] Using \( \sec \theta = \frac{1}{\cos \theta} \): \[ \frac{1}{\cos \theta} \times \sin \theta = \frac{\sin \theta}{\cos \theta} = \tan \theta. \] Thus, the correct answer is:
\[ \tan \theta. \]