Prove the following identities.
$\sec\theta=\frac{1}{\cos\theta}$

Verified step by step guidance

1. Start by recalling the definition of the secant function in trigonometry. The secant of an angle \( \theta \) is defined as the reciprocal of the cosine of that angle.
2. Express the secant function in terms of cosine: \( \sec\theta = \frac{1}{\cos\theta} \). This is the fundamental definition of the secant function.
3. To prove the identity \( \sec\theta = \frac{1}{\cos\theta} \), we need to show that this expression holds true for all values of \( \theta \) where \( \cos\theta \neq 0 \).
4. Consider the unit circle, where the cosine of an angle \( \theta \) is the x-coordinate of the point on the circle. The secant, being the reciprocal, represents the ratio of the hypotenuse to the adjacent side in a right triangle.
5. Since \( \sec\theta \) is defined as \( \frac{1}{\cos\theta} \), and this relationship is derived directly from the definitions of the trigonometric functions, the identity \( \sec\theta = \frac{1}{\cos\theta} \) is proven by definition.