Given that cos xo = $\frac{1}{r}$, express tan x in terms of r

Analyze the question: We are given cos x = 1/r and must write tan x using only r.

Concept: In a right-angled triangle, cos x = adjacent/hypotenuse, and tan x = opposite/adjacent. The missing side comes from Pythagoras' theorem.

Solution:

1. Since cos x = 1/r, take the adjacent side = 1 and the hypotenuse = r.

2. Use Pythagoras to find the opposite side: opposite² = hypotenuse² − adjacent² opposite² = r² − 1 opposite = √(r² − 1)

3. Now find tan x: tan x = opposite/adjacent = √(r² − 1) / 1 = √(r² − 1)

Check with the identity method: sin²x = 1 − cos²x = 1 − 1/r² = (r² − 1)/r² sin x = √(r² − 1)/r tan x = sin x / cos x = [√(r² − 1)/r] ÷ [1/r] = √(r² − 1) ✓

Tip: When you are given a trig ratio as a fraction, draw the right-angled triangle immediately and label the two known sides. Then use Pythagoras to get the third side. Remember that the hypotenuse is always the longest side, so when it is r you subtract (r² − 1), and when it is not, you add.