If sin x = $\frac{5}{13}$ and 0o $\leq$ x $\leq$ 90o, find the value of (cos x - tan x)

Analyze the question: We are given sin x = 5/13 with x between 0° and 90° (so x is in the first quadrant, and all ratios are positive). We need cos x − tan x.

Concept: Use the right-angled triangle with sin x = opposite/hypotenuse, and Pythagoras' theorem to find the missing side.

Solution:

1. Opposite = 5, hypotenuse = 13.
2. Adjacent = √(13² − 5²) = √(169 − 25) = √144 = 12.
3. cos x = 12/13
4. tan x = 5/12
5. cos x − tan x = 12/13 − 5/12

Find a common denominator (13 × 12 = 156):

12/13 = 144/156 and 5/12 = 65/156

So cos x − tan x = (144 − 65)/156 = 79/156

Answer: Option C, 79/156.

Tip: Whenever you see values like 5/13, remember the Pythagorean triple 5-12-13. Other common triples are 3-4-5 and 8-15-17, and recognising them saves time. Also, with fractions, always find the common denominator before subtracting. A common mistake is to subtract the numerators and denominators separately.