Question

Evaluate \( \frac{\sec(90^\circ - \theta) \csc\theta - \tan(90^\circ - \theta) \cot\theta + \cos^2 25^\circ + \cos^2 65^\circ}{3\tan 27^\circ \tan 63^\circ} \).

Hint:

Trigonometric Identities: Convert using \( \sec(90^\circ - \theta) = \csc \theta \) and \( \tan(90^\circ - \theta) = \cot \theta \).

Answer 1

Using trigonometric identities:
\[ \sec(90^\circ - \theta) = \csc\theta, \quad \tan(90^\circ - \theta) = \cot\theta \] \[ \Rightarrow \frac{\csc\theta \csc\theta - \cot\theta \cot\theta + \cos^2 25^\circ + \cos^2 65^\circ}{3\tan 27^\circ \tan 63^\circ} \] Since,
\[ \cos^2 25^\circ + \cos^2 65^\circ = 1 \] \[ \Rightarrow \frac{1}{3(1)} = \frac{1}{3} \] Thus, the value is \( \mathbf{\frac{1}{3}} \).
Correct Answer: \( \frac{1}{3} \)