Simplify tan(pi/6)-cos(pi/6)

Problem

tan(π/6)−cos(π/6)

Solution

1. Identify the exact value of tan(π/6) using the unit circle or trigonometric ratios.

tan(π/6)=1/√(,3)

2. Rationalize the denominator for tan(π/6) to make subtraction easier later.

1/√(,3)=√(,3)/3

3. Identify the exact value of cos(π/6) using the unit circle.

cos(π/6)=√(,3)/2

4. Substitute these values into the original expression.

√(,3)/3−√(,3)/2

5. Find a common denominator to combine the fractions. The least common multiple of 3 and 2 is 6

(2√(,3))/6−(3√(,3))/6

6. Subtract the numerators while keeping the common denominator.

(2√(,3)−3√(,3))/6=−√(,3)/6

Final Answer

tan(π/6)−cos(π/6)=−√(,3)/6

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