Simplify tan(105)

Problem

tan(105)

Solution

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

tan(105)=tan(60+45)

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(60+45)=(tan(60)+tan(45))/(1−tan(60)*tan(45))

3. Substitute the known values tan(60)=√(,3) and tan(45)=1

(√(,3)+1)/(1−√(,3)*(1))

4. Simplify the expression.

(√(,3)+1)/(1−√(,3))

5. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 1+√(,3)

((√(,3)+1)*(1+√(,3)))/((1−√(,3))*(1+√(,3)))

6. Expand the numerator and denominator.

(3+2√(,3)+1)/(1−3)

7. Combine like terms and divide.

(4+2√(,3))/(−2)

8. Divide each term in the numerator by −2

−2−√(,3)

Final Answer

tan(105)=−2−√(,3)

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