Simplify tan(75)

Problem

tan(75)

Solution

1. Identify the angle as a sum of two special angles from the unit circle.

tan(75)=tan(45+30)

2. Apply the formula for the tangent of a sum, which is tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B))

tan(45+30)=(tan(45)+tan(30))/(1−tan(45)*tan(30))

3. Substitute the known values tan(45)=1 and tan(30)=√(,3)/3

tan(75)=(1+√(,3)/3)/(1−(1)*(√(,3)/3))

4. Simplify the complex fraction by multiplying the numerator and denominator by 3

tan(75)=(3+√(,3))/(3−√(,3))

5. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 3+√(,3)

tan(75)=((3+√(,3))*(3+√(,3)))/((3−√(,3))*(3+√(,3)))

6. Expand the numerator and denominator.

tan(75)=(9+6√(,3)+3)/(9−3)

7. Combine like terms and simplify the fraction.

tan(75)=(12+6√(,3))/6

tan(75)=2+√(,3)

Final Answer

tan(75)=2+√(,3)

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