Question

Evaluate: \( \csc 39^\circ \cos 51^\circ + \tan 21^\circ \cot 69^\circ - \sec^2 21^\circ \)

Hint:

Use fundamental trigonometric identities to simplify expressions.

Answer 1

Using trigonometric identities:
\[ \csc 39^\circ = \frac{1}{\sin 39^\circ}, \quad \cos 51^\circ = \sin 39^\circ. \] \[ \csc 39^\circ \cos 51^\circ = \frac{1}{\sin 39^\circ} \times \sin 39^\circ = 1. \] \[ \tan 21^\circ \cot 69^\circ = \tan 21^\circ \times \frac{1}{\tan 69^\circ}. \] \[ = \tan 21^\circ \times \cot 69^\circ = 1. \] \[ \sec^2 21^\circ = 1 + \tan^2 21^\circ. \] Using approximate values:
\[ \sec^2 21^\circ \approx 1 + 0.14 = 1.14. \] Thus,
\[ 1 + 1 - 1.14 = 0.86. \]