If cos x = 4/5, 0° $\leq$ x $\leq$ 90°, find the value of (1 + tan x)/(1 - tan x)

Analyze the question: We are given cos x = 4/5 for x between 0° and 90°, and we need (1 + tan x)/(1 − tan x).

Concept: Use a right-angled triangle, where cos x = adjacent/hypotenuse and tan x = opposite/adjacent.

Solution:

1. Adjacent = 4 and hypotenuse = 5. By Pythagoras, opposite = √(5² − 4²) = √(25 − 16) = √9 = 3.
2. So tan x = 3/4.
3. Substitute: • Numerator: 1 + 3/4 = 7/4 • Denominator: 1 − 3/4 = 1/4
4. Divide: (7/4) ÷ (1/4) = 7

Answer: The value is 7, which is option B.

Tip: Whenever you see cos, sin, or tan given as a fraction, draw the 3-4-5 triangle (or another Pythagorean triple) straight away. It saves time. Also, when the fractions in a division share a denominator, like 7/4 ÷ 1/4, the denominators cancel, leaving 7/1.