For an acute angle A of a right triangle, explain why cos A is less than 1; sec A is greater than…

Step 1. (a) $\cos A = \dfrac{\text{adj}}{\text{hyp}}$. The hypotenuse is the longest side, so the fraction is less than $1$.

Step 2. (b) $\sec A = \dfrac{\text{hyp}}{\text{adj}}$ is the same fraction turned over, so it is greater than $1$.

Step 3. (c) $\tan A = \dfrac{\text{opp}}{\text{adj}}$ compares two shorter sides. Either may be longer, so $\tan A$ can be above or below $1$.